A Collection of Examples

D U B L I N U N I V E R S I T Y PR E S S S E RI E S .

A COLLECTION OF EX AMPLES

ON T HE

ANALYTIC GEOMETRY OF PLANE CONICS ;

T O W HICH AR E ADD E D

~ S OME E X A MPL E S ON S PHE R O CON JCS .

A RAL PH A . B E T M. O R S . R , ,

‘ S r M e z c l r e nio a t mm a i o D él . h T m z y C l leg e , u m

- E : H F co FT T T . DUBLIN ODGES , IGGIS , GRA ON S R E

- : LONGMANS co P T R O T ROW . LONDON , GREEN , , A E N S ER DUB LIN

PRINTED AT T HE UNIV ERSITY PR E SS ,

B Y PONSON B D EL DR I Y AN WV CK , P FA RE CE .

A PAR T of this collection of examples has been published by me before in a Collection of E xamp les and Pr oblems on. Comes and some o s f the Higher Plane Cur ves . In thi volume I

a x have dded a good many more e amples , besides giving solutions of the more difficult ones which were left nu

. I x or solved believe that either the e amples themselves ,

d of x . the metho s their solution , are to a great e tent original

A large number of the examples contain properties of

of circles connected with a conic, and especially those which I have double contact with the curve . n proving the pro perties of the latter systems of circles I have made frequent

of ff cc - use their di erential equations in elliptic ordinates, f the given curve being one of the system of con ocal conics .

In the same cc -ordinates I have also made use of the

f of to di ferential equations the tangents a conic, and the systems of conics having double contact with two ﬁxed

r ‘ f - con ocal conics . I he method of elliptic cc ordinates sim l p iﬁes greatly the study of relations involving the angles of i ff i intersect on of such systems , whose di erent al equations

s f take a imple orm . PREFACE .

I have added a section on Sphere-Comes at the end of

x the book . Most of these examples are e tensions of results already obtained for the case of the plane curves . I have

- again here made a free use of elliptic cc ordinates .

I have assumed the reader to be familiar with Dr .

’ S Conic S ections almon s , and have constantly made refe rencas throughout to that work . I have also occasionally

on Hi her Plane Curves referred to his works the g , and

eom tr o r i ns G e y f Th ee D me ions.

/i 1 88 . Mare , 4 CONTENTS .

S ECTION

INSCRIB ED TRIANGLES I ,

. CIRCUMSCRIB ED TRIANGLE S II ,

F - O A R A . SEL C NJUG TE T I NGLE S III ,

TRIANGLES FOR ME D B Y T W O TANGENTS AND THEIR C ORD IV . H CONT CT A , A O A R . CI R CL E S H V ING D OUB LE C NT CT W ITH THE CU V E V ,

CIRCLES CUTTING THE CURV E ORTHOGONALLY AT T O POINTS W ,

NORMALS ,

LINES MAKING A CONSTANT ANGLE IT T E CURV E W H H , OSCULATING CIRCLES ,

CONICS V ING DOUB LE CONTACT W ITH A FI X ED CONIC HA ,

RELATIONS OF A CIRCLE AND A CONIC ,

CIRCLES RELATED T o A ONIC C ,

RECIPROCAL TRIANGLES ,

MISCELLANEOU S EX AMPLE S

SP ERO -CONICS H ,

EXAMPLES ON CONI CS .

I — R BED A E . . INSC I TRI NGL S

1 . To ﬁnd the equation of the circle circumscribing a triangle inscribed in a conic . Let the equation of the conic be

2 Z $ y - — z b l —- U T- - O a g , Zi g and that of the Circle

z 2 ’ a + y — 2a a

if P of o of of U S then , Q are a pair ch rds intersection and , Z f S lL U AP we must have a relation of the orm Q , where Ii A B P x and are constants . ut if , Q be e pressed in terms of ’ of x S the eccentric angles their e tremities , we have ( almon s

Conics t 3 E Ar . 2 x . , 1 ,

— goos i h B) %sin §( a (3) OOS % ( a

S‘ . 1 “ 1 3 I 0 0 5 ” 3 - Sin -- -t 3 - COS - f Q 7 6 ) t(y ) §, (y Q Z

2 2 ffi Of x &c . hence , equating the coe cients , y , , in the identity

8 MU : AP Q , we get

2 2 a 12 Acos -g- (a L3)

2 2 — 6 Ii Asin -g( a B)

B 2 INSCRIBED TRIANGLES .

0 = sin (3)

f — 2aa - A{ coS % (a B) OOS - (y l

QOg/ = A{ cos g- (a B) sin §~ (5)

2 79 + - Acos % (a B) COS (6)

z F a — b rom we obt in , g 2 2 z ~1— la a sin a b cos of th and 2 ( B) § ( a B) the square e

semi a diameter parallel to P or Q ; and substituting these

V s of 8 A ii alue , , and in we get

‘M a + OS M 7 < o> (BM W AO ) , < )

a 3 sin - } sin a 8 ( £ ) tit 7 ) $47 ) , ( )

2 z —g- (a

2 (a (9)

Q S n of S [Z U if i ce the discriminant vanishes , must satisfy the equation

~ 2 2 2 a 5 [2 li

z z z z — a b - z t or a { P r mq b )

2 z z 2 N az i/ } a b r O ( 10)

2 ii h h r of 8 . Now W here is the radius the three values i , i Obtained from this equation are evidently the squares of the semi - diameters parallel to the sides of the triangle ; hence il /l i l Z a, of 10 r A if from the absolute term ( ) lso , we put

2 2 a if or 6 t in we can ﬁnd the expressions for

’ ’ a s of 15. , y from the ab olute term the equation in D INSCRIBE TRIANGLE-S . 3

We might ﬁnd the co-ordinates of the centre of S thus

E of S U liminating y between the equations and , we get

2 2 4 2 ’ 2 66 b w 4a w x & c O ( ) . ;

if x x x x f of hence , 1, 2, 3, 4 belong to the our points intersection

of S U and ,

2 2 a 6 93 + m $ 33 ( 1 g 3 4)

2 { cOS a cos B COS ) ! cos ( a B ( 11) 4a

is x which equivalent to the e pression already obtained . S b m imilarly, y eliminating , we get

2 2 6 66 { sin a + sin B + sin 7 ( 12) 4b

2 of . T o ﬁnd the equation the circle through the middle

points of the sides of the same triangle . Let the equation of the circle be

2 2 93 + y — 2x m

’ x cc - of of and let , g/ be the ordinates the centre the circum

a of b scribing circle , and , B those the centroid ; then , y a known geometrical relation

’ ’

2x 3a x 3 . , B y

Now we have

l a a + + b a + sin + sin a § (cos cos B cos y) , B % (sin B y) ; hence we have

2 2 (300 6 ) cos a cos cos cos a + + 1 ( B 7 ) ( B 7 ) , ( ) 8;

2 (b See (sin a + sin B + sin y) (2) 86 4 INSCRIBED TRIANGLES .

’2 To ﬁnd If we have k x y where r is the radius of the circumscribed circle hence

’ 2 2 k 1 { (3a x ) W W 2 }

' — — i— { 3a (3a 2x ) 3B(8B + y 7 }

2 2 “cos a cos B cos 7 ) { (a b )(cos a cos B cos 7 )

2 2 (a 6 ) cos (a + B

2 a + sin B sin 6 ) (sin a sin B sin 7 )

2 2 — (a — b ) sin

2 2 2 2 (a b ) { cos cos (B+ y) - (a + b )

2 2 2 M(a + cos Bcos 7 ) + (sin a sin B sin y) 1 }

2 2 — — ‘ — } (a 6 ) { 1 cos (a B) cos (B y) cos ( y a ) }

2 — - - 3 (o + o> ooo ( oz o) (a w oos ao o ) . < )

3 T o s of of . ﬁnd the locu the centroid an equilateral triangle inscribed in a conic . Equating the co-ordinates of the centroid and the centre of if a 8 the circumscribing circle , we get , B 7 ,

2 2 2 2 a ( a 6 ) cos 8 b(a 6 ) sin 8 . 2 2 2 2 a + 3b 6 + 3a hence the locus is the conic

2 2 2 5 a (a g( ( W .

2 We may ﬁnd the locus thus for the parabola 9 p x 0 . 2 2 2 Let x y 2a r 2By 76 0 be the equation of the cir cumscribin n n g circle , then eliminati g y between the equatio s 2 2 2 $ & c O of a . the circle and parabola, we get (p ) ,

$ m as l liminat~ whence 2 (2a p) 1 902 , , and similar y e SCRIBED T A G IN RI N LES . 5

or 0 . B ing , we ﬁnd yl yz y, m ut the centre of the of circle being the centroid the triangle , we have

3 a 2 0 50 3 y, B; hence 1 4, B g/4 ; therefore 2 96 Ma 220) f 4 . I the centroid of a triangle inscribed in a hyperbola f lies on the curve , the eccentric angles must satis y the equa 2 o a cos a sin sin 9 ti n (cos B cos y) (sin B , or cos g- (a B) cos g- (B 7 ) cos % (y a ) 1 (1) hence from ’2 2 2 2 a 6 Ex . k we have , from which we see that the circle passing through the middle points of the sides of the triangle cuts orthogonally the director circle . The eccentric

‘ of angles are course imaginary for the hyperbola, but this ff does not a ect the validity of the proof . When the rela

1 ﬁ of a f tion ( ) is satis ed , the area the tri ngle ormed by the tangents at the v ertices is equal to half the area of the given triangle . The relation (1) can also be written in the form

oh in o oin 6 o J ns 7 ) m (7 ) J o 1 ) , from which it can be seen that the ellipse touching the sides of the triangle at their middle points passes through the

centre of the curv e .

’ 2 2 — 2 — 2 5 . I f x + y x x /z y + O (1) represents the circle passing through the extremities of three

- of the 1 0 semi diameters ellipse , show that $5

’ 2 2 s x y (a 6 i ) o (2)

represents the circle passing through the extremities of the

- three conjugate semi diameters . 6 INSCRIBED TRIANGLES .

Also show that if the circle (1) cuts orthogonally the circle

z 2 — 2 m + 2ax + 75 0 y , the circle (2) cuts orthogonally

2b 2 2 2 g- — x + y + Bw ay + a + b g a

6 C f . orresponding points on the con ocal conics

2 2 x y 90 y + l — U — O + l — V — O 2 2 , ’2 ’2 a b a b are connected by the relations

’ ’ at: cc y y . / ’ l 7 a b b

2 2 2 show that if x y a 2 By k 0 represents the circle

on U passing through three points ,

2 2 ’2 — By + k + a — a = o will represent the circle passing through the three corre sponding points on V.

7 of of . To ﬁnd the locus the centroid a triangle inscribed 2 2 90 3/ l n one m 1 E U O i i co e 2 2 , and c rcumscr bed about a 6 another whose tangential equation is

2 E A B 0 F G H A v E 0 . ( , , , , , ) ( , p , )

of U Writing down the condition that the chord ,

Q7 — - (eu r o) am m oos aa o w

2 for s should touch , and two similar equations the other side of u n b sin a sin the triangle , m ltiplyi g them y ( B) , (B INSCRIBED TRIANGLES . 7

sin a (y ) , respectively, and adding them together we get , 1 1 f b sin a sin —— a ter dividing y ( B) , 2 (B y) , sin 3, (y

— 1 1 C 1 4 cos a cos - a { g( B) 9 B 7 ) cos 9 (y ) } ( Y — 2

— 2 si n a sm s1n O . ( 1 E( B 7 ) )

B ut Oos a cos B cos 7 sin a sin B Sin 'y 3 and 1 + 8 cos % (a

from we have for the equation of the locus the

" - O 2 + . ( ) b

of There is , course , an invariant relation connecting 4 E U if H 0 n and ; and we have also , , it can be show 22 if

F r if of t x . o c that the centroid the riangle is ﬁ ed , we limi nate 7 between the equations

~ cos a + cos B + cos 7 sin a + sin B + sin y

2 l— 1 + 4 — — 12 — cos - a — cos cos §(a B) z Q ( B) g

2 — — - 12 oos a sin a + o é ( B) g, ( c) ,

93/ 2- 12 o 1 12 v A+ . 62 2

Putting 0 O in we see that the locus of the cen troid of a triangle circumscribed about a conic and inscribed 8 INSCRIBED TRIANGLES .

ma in a parabola is a right line . This result y be readily arrived at geometrically by projecting the conic orthogonally f into a circle . The centre o the circumscribed circle is then

x of s direc ﬁ ed, and the intersection perpendiculars lie on the & x of t f c . tri the parabola here ore , f 8 . Since the centroid of a triangle is the pole o the line at w b inﬁnity ith regard to the triangle , y projecting the

of x see if results the preceding e ample we that , a triangle be U V inscribed in a conic , and circumscribed about a conic ,

u of x L the loc s of the pole a ﬁ ed line , with regard to the n u if L V triangle , is a co ic , which red ces to a line touches , if U L f L . A V the other actor being lso , , , and are connected bytwo relations besides the invariant relation between U and x L is V of L w to . , the pole , ith regard the triangle , is ﬁ ed

f of of tw o of U w hich then , in act , the chord contact tangents

c tou h V. 9 A ’ s x . B C , , are the vertice of a ﬁ ed triangle inscribed

P on : in a conic , and is a variable point the curve to ﬁnd the locus of the centroid of the triangle formed by the lines

PA PB P C . bisecting , , , at angles 2 g right 3—— L et be 1 : O a eccen the conic , and , B , 7 , (p the ; 22

' of A B C P tric angles , , , , respectively then the intersection

x of n PA PB ( 1 , y,) the li es bisecting , at right angles is the

of P A B f centre the circle passing through , , ; hence , rom 2 11 1 E x . 1 h ( ) and ( ) , we ave

2 6 — + in + sin < sin a + ( { sin a s B p ( B M) , 4b

e for cc - n ( of er and similar valu s the ordinates a g/ 2, my, the oth

e vertices of the variable triangl .

10 INSCRIBED TRIANGLES .

’ 2 ’ 2 2 ’ Now when a circle (90 x ) (y y) r S O is in i 2 s m 111 the 4- 1 5 S : 0 cribed a triangle inscribed conic , ;2 2:

’ we must have the relati on 4 A 6 0 between the inva ’ S 0 0 72508 Ar t 3 E 7 x . . 1 riants, or ( almon s , ,

2 2 2 1 - r (61 as y

Solving this equation for r we get

2 2 2 ’ e 0 ( 1 6 x wh re ; hence , putting

64 2 E

’ subject to which conditions it can easily be seen that S has double contact with the conic

2 x y 2519

2 6 O

11 . Triangles are inscribed in a conic and circumscribed about a ﬁxed circle to ﬁnd the envelope of the cir cumscrib

l s ing cir c e .

L A B 0 a et , , be the vertices of the triangle , and , B, 7 A B 0 o the perpendiculars from , , n a line drawn through the centre of the ﬁxed circle parallel to one of the axes of the

‘ AB = ' 0A . . B 0 . a 1 is O B a curve , then B y ut , B , y are ’ 15 15 15 A B O proportional to the tangents 1 , 2, 3 drawn from , , to the circle having double contact with the curve at the points ’ b x S (Ionics where it is met y the parallel to the a is ( almon s , ‘ 2 Ar 261 1 B O . t 35 0 A . z. 1 AB t O t. ) hence from ( ) , 2 , ( ) INSCRIBED TRIANGLES . 1 1

’

Now b Dr . C em f , y asey s theor , if our circles are all touched b b &c . y a ﬁfth , their common tangents are connected y ’ the relation (12) (34) i (18) (24) j: (23 ( 14) 0 (Salmon s

Comics Ar 12 a of t. 1 b , ( )) hence , y supposing the radii the

3 see 2 ex circles ( ) to vanish , we that the relation ( ) presses that the circle hav ing double contact with the curve A B S ma touches the circle passing through , , ince we y draw a parallel to either axis through the centre of the ﬁxed

see l h circle , we that the circumscribing circle wil touc the two circles having double contact with the curve at the points

where it is met by these parallels .

r 12 . A triangle is inscribed in a conic so that the ci cum scribing cir cle touches a ﬁx ed circle having double contact with the conic ; show that one of the circles touching the sides has double contact with a ﬁxed parallel curve to a

E x . 1 1 E 10 conic (see and x .

l - 13 . A tr ian e f c V g , sel conjugate with regard to a coni , is inscribed in a conic U ; show that circles having double con

U of V out tact with , whose chords contact touch , orthogonally th edirector circle of V.

4 G e r 1 . if f l e iv n a t iangle inscribed in a conic, a ocus i s

on xe of r a ﬁ d circle , to ﬁnd the envelope the cor esponding

directrix .

I f a f p l , pg , p 3 denote the dist nces of a point rom the ver

of r a b c can f m tices the t iangle , and , , the sides, it be seen ro ’ Ptolemy s theorem that

_ 2 2 2 2 2 2 2 2 2 2 2 b 0 2 0 a 0 2 66 0 0 0 5 U p 2 p3 P3 1 1 1 31 5

is proportional to the square of the tangent from the point to

f an the circumscribing circle . It ollows then that y circle can 2 2 2 be e the f U m n 0 B writt n in orm pg p3 ) ut f p l , p g, p g are proportional to the perpendiculars rom the ver 12 INSCRIBED TRIANGLES . tices of the triangle on the directrix ; hence from (1) we have for the tangential equation of the envelope of the directrix

4 4 4 4 4 4 2 2 2 2 z z z z — 2 2 2 2 d a 6 8 0 7 26 0 6 7 Q a a y a 26Z 5 a 6 2 2 2 2 0 ( la mB ny ) .

4 4 Since a a & c . is the product of the factors

— — — aa + b + c aa + b — C aa - b t w aa b C B y, B y, B y, B y, it follows that the envelope has the centres of the circles touching the sides for double points .

15 . L AB C U 2 et be a triangle inscribed in a conic , and a circle having double contact with U at points on a parallel ’ its x L t t of to minor a is . et , be the lengths the direct and transverse common tangents of 2 and the circle cir cumscrib

AB C 9 ing , and let p l , p g , p g, 1 4 be the perpendiculars from the f B C CA AB on centres of the our circles touching the sides , , 2 ’2 4 of o U S 15 t 6 9 9 0 9 2 . the chord contact f and how that 1 11 21 31 4,

e f U E x where is the eccentricity o (see .

16 . A triangle is inscribed in a variable parabola ; to ﬁnd the locus of the intersection of the perpendiculars of the tri angle formed by the tangents to the curve at the vertices of

the triangle . Consider two consecutive parabolas indeﬁnitely near one o u an ther, circ mscribed about the triangle , then the intersections of the perpendiculars of the triangles formed by the tangents

of c lie on the directrices the orresponding parabolas , and ,

therefore , the ultimate intersection of the directrices is the

point whose locus we seek . Thus we see that the locus coin f cides with the envelope o the directrix .

L et of pl , pg, p, denote the distances a point from the

v the a b c A B C s ertices of triangle , and , , , , , the side and INSCRIBED TRIANGLES . 13

f angles, respectively, then we can veri y the following iden tical relation

2 2 2 260 cos A 200 B pg p3 cos p3 p1

2 2 2 2 2040 C 20 60 0 6 cos p 1 p 3 ( cos A p 1 cos B p 2

2 2 2 + 0 c b 1 os 3 ) + a ( )

B ut a f p l , pg , p, are equal to the perpendiculars , B , y rom the vertices on the directrix ; hence from (1)

2 4 2 4 2 2 2 2 2 2 a a b B 0 7 250 cos A B 20 0 cos B 3 01

2 2 2 2 20 6 cos C a B 20 00 (0 cos A a 0 cos B B

2 2 2 2 0 C 0 6 0 0 cos 7 ) , which being rendered homogeneous bymeans of the identical

e a v iz . r lation connecting , B , y, ,

2 2 2 2 a a b B 0 7 260 cos A By 200 cos B ya

2 20 0 cos C a 4 A B ,

shows that the envelope is of the fourth class .

Fo It can be shown that the curve (2) is universal . r

l ; m n 0 the tangential equation / 4/ B f , where Z m n 0 represents a series of parabolas circumscribing

cc - of x the triangle , and the tangential ordinates the directri

x as f of l m n of can then be e pressed unctions , , the fourth

’ ' Com s Ar S 0 t. degree ( almon s , The locus thus appears to be of the sixth degree .

- f of A 17 . cc t o B C The ordina es the centroid a triangle ,

2 a: 0 a inscribed in the parabola g/ p , are , B show that the cc - 00 of of f b ordinates , y the centroid the triangle ormed ythe e A B C b tang nts at , , are given y the equations

2 — 150 3 a . 27 B p , 3/ B T 14 INSCRIBED RIANGLES .

18 . Show that the equation of the circle passing through

of of the middle points the sides a triangle , inscribed in the 2 90 0 parabola y p , is

2 2 2 2 2 4 9 so 2 9 3 9 w 2 2 — 1 ( 0 ) 0 (7 1 0 2 2 ( 0 10 1 p op o p o) y

2 2 9 9 + 9 ” 0 2 2 (2 2 2 ) ,

9 sum of where p l , 1 2, p , are the sum , the products in pairs ,

of of and product , respectively, the ordinates g/ 1 ya, y, the f vertices o the triangle . A 19 . triangle AB C is inscribed in a parabola whose focus is F ; to show that one of the circles touching the lines FA FB FC s which bisect , , at right angles pa ses through the AB centre of the circle circumscribing C.

L et 72 of P f ph 1 , denote the distances a point rom

F A B C if a er endi , , , respectively; then , B , y are the p p

culars f P FA FB F C rom on the lines bisecting , at right

angles, we have

2 2 2 70 2 01 01 2 d 2 01 1 p 1 1 , p g B, p 37 , ( )

& ow d FA c . N where l , the equation of a circle touching " a , B , y is

— - cos g— A / a cos 5B / B cos é C / y 0 ; or from ( 1)

— sin 0 0 sin ;P(02 03) <5 ( 3 1)

/ 2 — P + sin 0 : é 2) j w 0 0 0 FA FB F C a here 1 , 2 , 3 are the angles which , , m ke with the axis of the curve . INSCRIBED TRIANGLES . 15

B of A B C ut for the centre the circle passing through , , , r r 7 f th e of , , 3 , and rom polar equation the parabola

2 2 2 d m seo ~ 0 d m sec 0 m sec - 9 l g 1, z % 2 , g

2 & f r f c . which values satis ythe equation ( ) identically; the e ore , 2

20 is s 1 0 if . A triangle in cribed in the conic ; x o of of s , y are the c ordinates the centre the circum cribing

ad z of circle , and , / those the centroid , show that

2 2 4 2 ’ 4 16 ( 00 90 9 0 12 0 yy) 0 0 .

s 21 . A triangle is inscribed in a conic o that the circum

x one of i scribing circle passes through two ﬁ ed points , wh ch is on the curve ; show that the centroid and intersection of the perpendiculars lie on ﬁxed lines .

S a - how also , in the same c se , that the nine point and polar

s circles have double contact with ﬁxed conic .

22 A U so . triangle is inscribed in a conic that two sides are parallel to ﬁxed lines show that the locus of the centre of the circumscribing circle is a conic having the same centre and axes as U. f f 23 . I the centroid o a triangle inscribed in a conic lies on a concentric, similar and similarly situated conic, show that the nine-point circle cuts orthogonally a ﬁxed circle S f concentric with the curve . how also that the area o the triangle is in a constant ratio to the area of the triangle formed by the tangents at the vertices . f 24 . A B C t , , are three poin s orming a triangle inscribed in so v x a conic, that the tangent at each erte is parallel to the I f s P an on . PA PB oppo ite side and is y point the curve , ,

PC s of i e L M N meet the opposite side the tr angl in , , , show B T 16 INSCRI ED RIANGLES . that the area of the triangle L MN is double that of the triangle AB C. 25 An x . ellipse circumscribes a ﬁ ed triangle so that two of the vertices are at the extremities of a pair of conjugate diameters ; show that the locus of its centre is a hyperbola f - with regard to which the given triangle is sel conjugate . 6 A x A if 2 . conic circumscribes a ﬁ ed triangle B C ; the

of AB s diameter the conic parallel to be given in length, how

is that the locus of its centre a conic, whose asymptotes are A C B C C parallel to , , and with regard to which is the pole

of AB .

2 A e x n of 7 . conic circumscrib s a ﬁ ed tria gle so that one the vertices of the triangle is a vertex of the curve ; show that the tri linear equation of the locus of the centre of the conic is a ( a B cos C) (B sin B + y sin C a sin A)

— a C si A sin B sin C 0 B(B cos ) ( u n B 7 ) .

2 A 8 ’ AB C 8 . circle circumscribes a triangle inscribed in 2 ’ S if 9 4 A9 0 of a conic , show that the algebraic sum

of 8 AB B C CA z the diameters parallel to , , is equal to ero , ’ 4AG = 0 sum of and if , show that the algebraic the ’ is u z A 6 A reciprocals of the diameters eq al to ero , where , , n ’ ’ ' 1 a of S 8 see S Comos Art. 37 are the i v riants and ( almon s , , E x .

18 CIRCUMSCRIBED TRIANGLES .

2 2 ’ 2 L 66 5 20 50 70 9 3 where ( ) cos 1 , ( )

2 2 2 M 0 5 sin ) 207 k 4 ( ) 4 / q, ( )

2 2 — ’ — ’ N : a + b 2pax 2g oye + g (5)

B ut 2 b a r es ec since ( ) is true when y is replaced y and B , p 0 tiv el u L M N . E y, we m st have liminating, then , f g/ rom these equations , we get

2 2 2 2 { a b 5 ) (p cos q) q 2 2 13 Q — 1

2 2 2 c + b + (a

4 0 0 8 ( a L3) 0 0 8 (L3 7 ) cos t al l since

2 2 — — — — 2 and p + 9 0 0 8 (a ﬁl (B r ) OOS % (r - a ) }

— — (13 7 ) cos t- (r a A

H ma of 8 ence , ﬁnally, we y write the equation thus

2 2 2 2 2 x + y %{ k (cos a + cos B+ cos y) ( 0 0 ) cos ( a + B + y) }

2 2 — 2 { lo (sin a sin B + sin y) (a b ) sin (a + B + y) }

2 19 0 (7) 2 where [6 has the value We may also ﬁnd the equation of the circle as follows

I R is of A A B C f the radius the circle , and , , , the area and

of angles the triangle, respectively, we have

’ 2 sin A simB sin C CIRCUMSCRIBED TRIANGLES . 19

A = aotan —5(a B) tan % (B 7 ) tan % (y 01)

’

3 x . S Combs Art. 2 1 E ( almon s , ,

l A j and sin 2 2 i 2 7/ (66 sin B b ijsiiiﬁf siz 7 + 5 cos 7 ) }

& sin B c . hence we have

2 2 2 2 2 2 2 2 2 2 2 a sin a 5 cos a ) (c sin B 5 cos B)(a sin y 5 cos y)}

4 040 cos g—(a B) cos 5- (B 7 ) cos 5 a ) (8)

’ ' ’ Now a a f the let , B , y, , B , y be the perpendiculars rom ’ ’ 2 f of o u 6 oci on the sides the triangle , then BB ,

’ ’ ’ and a sin A B sin B 7 sin C identically therefore

2 o (By sin A ya sin B a B sin C)

o f = but By sin A ya sin B a B sin C sin A sin B sin Ct ’ 2 Combs r 132 E x . S A t. ( almon s , , )

2 2 hence 0 2 2R a By ; (9)

’ ’ ' of s 2B a 10 and , cour e , also B y ( ) u for the other foc s . 2 ’ 2 ow lo 2050 0 75 N we have ,

2 ’ k + 2cx +

x a of and if we e press , B , 7 in terms the eccentric angles , we get 0 (0 cos a a) 2 2 2 a sin a + 6 cos a 20 CIRCUMSCRIBED TRIANGLES . hence (9) and ( 10) become

— — a — 0 c a 0 cos c 0 2 ( cos ) ( B) ( cos 7 ) 15 cc + 0 — — 2c cos -- a COS z a 2 ( B) y) g )

a 0 cos a 0 0 as 0 ’ ( ) ( cos B) ( cos y) 2 2 2 7L 2050 0 ’ 2 — — - — — 54 cos g( a B) oos {5 (B y) cos Hy a ) which give by subtraction

2 61 (cos a cos B cos y) 0 cos a cos B cos y) 3 4a cos - (a B) cos — (B y) cos é— (y a)

B which is equivalent to the value already obtained . y sym

or b of f metry, then , y means the imaginary oci , we arrive at

2 2 5 (sin a sin B sin 7 ) 0 sin a sin B sin 27

40 cos g- (a B) cos — (B 7 ) cos -g— (y a ) Also adding the equations (11) and (12) we get

2 2 " a 0 (cos a cos B + cos B cos y cos y cos a )

— — 4 Z oos —o( o 4 5) oos s(0 w ) ooo s a ) From (9) and (10) we get at once

2 2 2 2 25 25 4 5 12 ,

the invariant relation connecting the circle and the conic . We can obtain an expression for the radius of the circle ’ ’ ’ of of f in terms the distances p l , pg, p g, p l , pz , p3 the foci rom b’ ’ of n . L A the vertices the tria gle et , , be the sides and area respectively of the triangle formed by the feet of the perpendiculars fr om a focus on the sides of the given triangle ;

x of c then , since the au iliary circle the coni passes through f the feet o the perpendiculars , we have

’ ’ 0 0 4 0 A ; CIRCUMSCRIBED TRIANGLES . 2 1

’ ’ A b B 0 C , pg sin , p3 sin ,

’ 2 2 A sin A sin B sin 0 6

’

S Combs Art. ( almon s ,

2 2 0 25 Hence P1 pg p a ,

for f s and similarly the other ocu ,

therefore from (15)

0 1 0 2 9 3 R2 1 1 2 2 16 a 5

To ﬁnd the equation of the polar of the triangle by three tangents to the conic

2 2 x y - 1 0 . 2 2 0 5 " 2 2 L et the equation of the circle be (x (y y ) 72 0 ’ ’ w cc- s of of i and let , y be the ordinate the centre the c rcum

b a b scri ing circle, and , B those of the centroid ; then , y a

known geometrical relation , we have

” " : x 3a y 3B 2g/ .

1 3 - - a a cos é ( a + B) cos 9 (B 7 ) cos 5 (7 )

- - —- — — o coo o(ow 16 > ooo o(o v) (7 o >

3 (cos a + cos B + cos 7 ) + 4 cOS a cos B cos x

-— — — — 4 ooo 1 (o c) ooo -s- (o s oos oo o )

’ u 2 9 50 Ex . hence , substit ting for from we get

2 2 2 a { cos a cos B cos 7 cos ( a B 2 (66 4- 5 ) COS a COS B 0 0 3 7

— — — 2 — — 4 a cos % (a B) cos é (B y) cos é (y a) l and, simi arly,

2 2 2 — o sin a sin sin sin 2 c 6 sin a sin sin { + B 7 + (a. B ( ) B y

4 b cos é ( a - B) a ) 22 CIRCUMSCRIBED TRIANGLES .

2 2 Also the absolute term so y 7° 05 6 ; for by the invariants we know that this circle cuts orthogonally the

’

s 3 5 E x . S Comb Ar t. 7 director circle ( almon s , ,

72 25 t t of I f be the radius of this circle , and 1 , 2 , 3 the lengths the tangents from the vertices of the triangle to the director circle , we have 2 A cot A cot B cot C ;

cot A cot B & c . 2 aotang- (B ’ 69 C mbs Ar . 1 x 3 S o t E . ( almon s , , )

—- — — — — A abtan é(B 'y) tan g(y a ) tan ( a B) ;

2 2 2 $1 $2 75 2 3 r therefore 2 2 4 a o

For if r one of 25 1 the ellipse , is real , the quantities 1 , a, 2,

of . course , imaginary

2 For 4 mm 0 the parabola , y , we ﬁnd

9 9 29 2 1 2 2 1 3 m

f f where p g are the perpendiculars rom the vertices o the triangle on the directrix . 31 f b . The circle circumscribing the triangle ormed ythree tangents to a conic passes through the centre of the curve ; show that three extremities of the diameters parallel to the tangents lie on a circle passing through the centre of the curve .

32 the - f f b . If nine point circle o the triangle ormed y

three tangents to a hyperbola 1 0 passes through 22 i,

c of n of the entre the curve , show that the ce tre the circum 9 scribing circle lies on the confocal hyperbola ; Z, CIRCUMSCRIBED TRIANGLES . 23

33 S of f b . how that the polar circle the triangle ormed y three tangents to an equilateral hyperbola touches the nine point circle of the triangle formed bythe points of contact at f the centre o the curve . 3 4 . Given a triangle circumscribed about a conic and the

of x of f length the a is maj or, show that the locus the oci is a

of x of of curve the si th order, which the vertices the triangles 9 are nodes (see E x . 2 35 . To ﬁnd the locus of the centre of the circumscribing

of a o S in circle a triangle circumscribed about c nic , and ’ scribed in a conic S

I ma b t y be shown y the invariants that, if a triangle be S S ’ circumscribed about a conic , and inscribed in a conic , it

- is also self conjugate with regard to another ﬁxed conic . ’ ’ f bs Ar 376 or S S S Com t. Taking the values and in almon s , , we ﬁnd the following value for the covariant F :

— 2 2 2 5 4 1350 ﬁ e + z 4 + 13 s z x hw (9 f/ j y ) (f g ) (f y g y) ,

2 2 2 f 050 h s 0 ex rom which it is evident that 9 fy + f y , being ’ f x 2F GS 0 . pressible in the orm , represents a ﬁ ed conic Let us put 2 2 2 2 0 _ — A 2 8 1 ; a b c 12 50 S , a af b 9 3 , y? U ; d 9 ( ) ( then if we substitute the cc-ordinates of the intersection of the tangents

x — + — -- - in l r - O l OOS a Sin a l 0 cos l s , ) % , B g B (

S 0 in , we get

’ 2 ’ 2 b b cos 01 cos B (b b sin a sin B

’ 2 ’ ’ b b 0 sin (a B) 2 g 0 (cos 0 cos B) ’ 0 2f b (sin a sin B) . T 24 CIRCUMSCRIBED RIANGLES .

But this relation evidently expresses that the tangents (1) ar e conj ugate with respect to the conic whose tangential equation is

2 2 2 2 2 2 0 679 A 6 0 0 u v ,

/ , ' f z z z l it a b Aju 4g co s 4f b uv 1 0 (2)

’ and triangles circumscribed about S and inscribed in S are

- f w 2 . Now e then evidently sel conjugate ith regard to , sinc the circumscribed circle of the triangle cuts orthogonally the

of 2 director circle , we have

2 ’ 2 2 2 2 ' 2 ’ 2 (0 0 b b (00 y 72 ) 4(g 0 oo f b yfe)

2 2 ’ 2 2 0 6 0 0 0 0 ( ) ( ) ,

0 cc - of of the where , y are the ordinates the centre circum " ombs 94 7 A 2 E x . C rt. scribing circle , and is its radius ( , , ) 2 ’2 2 2 4 12 f E x . 29 15 t é 6 or We have also rom ( ) ,

2 2 2 2 2 (50 y 0 5 620

92 3 hence , eliminating between ( ) and we obtain

’ 2 2 2 ’ 2f b y0 0 0 (0

’ 2 2 2 2 2 c ) (a w 0 6 ) (5)

’ 2 ’ I b b 0 0 of f , this conic becomes the square a line, and in no other case will its discriminant vanish ; for, if

of f b we form the discriminant we get , a ter dividing y b’ b2

2 2 ’2 ” 2 2 ’ ’ 2 079 40 45 0 6 0 b g 7 ( ) , and this combined with the invariant relation connecting ' S z S v i . and , ,

2 2 2 2 ’2 2 ’2 079 40 0 15 ) 40 2 g 4bf

' ’ 2 ’ ’ ’ ’ 0 b 0 0 b 0 0 o S be gives ( ) , whence , , r must

Q6 CIRCUMSCRIBED TRIANGLES .

f of S inscribed in another conic passing through the oci , the circumscribing circle will touch two ﬁxed circles also passing I f 3 5 f of S . S through the oci becomes a circle , ( ) and ( ) have double contact with each other , and the envelope

of breaks up into two circles , which agrees with the result E 1 1 . x . 3 f 7 . To ﬁnd the locus of the centre o a circle which touches the sides of a triangle circumscribed about a conic S S ’ , and inscribed in a conic ” ” ” ” L b is 50 E V = 0 et , f g , ) ( , y, be the equation

0 cc - of 2 in , y ordinates the conic which we have called in

E x 35 2 if 0 - f of . cc o ( ) then , , y are the ordinates the centre

77 the circle , and its radius , we have

V : + 0 ) since the circle touches the sides of a triangle self-conjugate

Combs A B rt. with regard to V ( , ut we have also the ’

n t S v iz . invariant relation connecti g the circle wi h the conic ,

2 2

E 10 3 if S / f x . ( ) be written in the orm

i r is of . hence , elim nating , the locus a curve the fourth order I S ’ f the conic become a circle , the locus is a bicircular quartic ; for if we put

2 2 ’ 00 1 S V : 0 5 0 70 y , and ( , , g, ) ( , y,

’ 2 2 w e h f E 70 1 w ave , rom uler s equation , y hich , com

2 2 2 2 : 4 bined with V 7 (0 gives (0 b) (0 l ) V. U E CIRC MSCRIBED TRIANGL S . 2 7

n e This quartic , whe certain conditions are satisﬁed, can br ak P 7 . 7 0 up into two circles utting , and equating 4 V — d + é

2 2 2 ’ of 50 + 750 17 x 970 is to the product the circles y 1 , 7 we get conditions which , combined with the invariant relation 2 2 ’ 0 0 b n V S f ( ) f g co necting and , give the urther 2 2 2 0 0 6 . m of relations f , 9 The latter relations ight, b 0 2 2 2 0 0 . course , be replaced y g , f in 38 . A triangle is circumscribed about a conic S and ’ in S n scribed a conic , havi g double contact with the director circle of S show that the polar circle of the triangle touches two ﬁxed circles . I f S ’ is a parabola, show that the polar circles form a x coa ial system .

39 f - . To ﬁnd the locus of the centre o the nine point circle of a triangle circumscribed about a conic S and inscribed in a S ’ conic . 2 2 2 Let 0 62

' 2 of S A B C F G H A v 0 be the equation , and ( , , , , , ) ( , u, ) ,

t of S x the angential equation then , e pressing that the chord

( a S , we get

cos2 2

0 ,

29 ma f E x . which, as at y be written in the orm 28 CIRCUMSCRIBED TRIANGLES . where 2 7 7 + (2) i f. 5

cos (I) (3)

QF sin < ) cos (p (4) 1 b

B ut 1 b a since ( ) is true , when 7 is replaced y , B , respectively,

- L M N 0 . Now 00 cc we must have , if , y are the ordi

of of the - o nates the centre nine p int circle, we have , from E 2 1 x . , ( ) and

2 2 2 2 2 2 80 7: 30 b 0 ( 86 0 36 0 sin ( )p cos p, 3/ ( ) q ¢ ;

M 0 x hence, from N , we get e pressions of the form

’ u s sin g , f rom which it follows that the locus is a conic . ’ ' I f u i v v 1? u , , , , this conic will become a circle , and

A B C (0 i b) then we have 2 2 0 0 0 0 I f ’ ’ v v O . u u , the locus reduces to a line

40 . A triangle is circumscribed about a conic S and in ’ scribed in a conic S ; show that the nine -point circle cuts x orthogonally a ﬁ ed circle . 1 4 . A triangle is circumscribed about a conic S and in ’ scribed in a conic S show that the radical axis of the

- x circumscribing and nine point circles touches a ﬁ ed conic . 42 A . triangle is circumscribed about the comc

2 2 x y + 2 2 0 b

2 2 2 2 and inscribed in the circle (0 a ) (/7 B) 7 0 ; if s be D CIRCUMSCRIBE TRIANGLES . 29

half the sum of the angles subtended by the foci of the conic

of at the vertices the triangle , to show that

2 2 2 2 2 " 0 7 0 2 ' 0 7 3 0 0 7 s . B cos , B sin E 29 x . 9 f The relation at ( ) is , of course , also true or the f x 6 imaginary oci, with the e ception that must be replaced by

2 2 2 0 0 25 27 0 . Now 1f 50 0 0 8 0) ; thus we have By , y sin w

0 of the p be the equation a tangent to conic, we have —

a 1 sin w but 0 0 0 ) 0 ; p cos ¢ , sin sin ¢ , where (p is the angle the focal radius vector makes with the normal ;

” - 2 1 a 0 0 for . N can hence, and similar values B , y ow it be shown by geometrical considerations that the sum of the angles subtended by the foci at the points of contact is equal to the sum of the angles subtended at the vertices of the n 02 f tria gle hence, w— e have there ore , putting 2 2 l 25 a 0 i T (B M Y, and equating real and imaginary

s . part , we obtain the results required

43 of . To ﬁnd the locus of the centroid an equilateral tr l iang e circumscribed about a conic . w L t -1 1 = 0 e be the equation of the conic ; then , ga from the invariant relation connecting the conic with the cir cumscri n f Ex 29 bi of . g circle the triangle , we have , rom R 4 (1)

R of 0 cc- where is the radius the circle, and , y the ordinates of i s t . A centre gain , from the invariant relation connecting the conic with the polar circle, we have (2) where M is the rectangle under the segments of the perpen M 122 dicular s . B ut, for an equilateral triangle , i ; hence, eliminating R between the equations we get

2 2 2 2 (350 0 b y 30 CIRCUMSCRIBED TRIANGLES .

44 . To ﬁnd the equations of the circles touching the sides of a triangle circumscribed about a conic . / L et of be S : 0 the equation the conic 2 , and z6 6

2 2 — ’ “2 =— that of one of the circles (so (g/ 30 7 S 0 ; then the conditions that the line A02 my 1 = 0 should touch S and S ’ are

2 2 2 2 + 6 0 0 A M , respectively.

ow if r of 2 — N , we form the disc iminant we get gé

0 7 — (1) 2 9— 2 3— 2 2 0 17 19 17 72

which is identical with the equation alreadyobtained in E x . 1

’ 2 ’ for of S 0 S . B 2 2 the discriminant ut , writing g: in

2 — 2 2 2 2 2 1 ’ (0 h )A (6 17 111 1 (70 A we see that the form £,

2 ’ E — 2 when represents two points, these points must lie 52, 2 2 2 2 2 on f 0 2 0 A 0 0 1 O the con ocal conic ( ) ( ) u ,

2 2

' ' z l l o' 2 z z g z ( ) at b h

Hence it follows at once that the extremities of a diagonal of ’ the quadrilateral formed by the common tangents of S and S

on s f 2 lie the ame con ocal conic, three conics such as ( ) corre sponding to the roots of the equation From if 1) be half the major axis of a confocal conic passing through a 2 2 2 x of 0 0 v verte the triangle , we have ; and the equa tion (1) then becomes

2 v 0 CIRCUMSCRIBED TRIANGLES . 31

3 semiaxes v 17 11 of This equation ( ) gives the 1, 2, 3 three confocal conics passing through the vertices of the triangle ; and from its absolute term we get

v V U 4 1 2 3 , ( ) and similarly from the absolute term of the equation whose roots are

2 2 2 2 — 2 ’ 2 2 2 v 0 17 0 v bcz 0 17 0 v 11 1 , 2 , 3 , y ( 1 ) ( 3

4 Also fromthe coefﬁcient of v we have

’2 50 + y hence the equation

2 12 1 V Q V 3 53 2 2 2 2 2 2 x/ (0 v l (c ” 2 1 3 60 ) 0

2 — 0 O (6)

represents one of the circles touching the sides of the triangle . I f S confocals ae for is an ellipse , these three are hyperbol the

f 2 s miax s . I 1 1 e e circle inscribed in the triangle 1 1 , M2 , 1 3 are the of the three confocal ellipses passing through the vertices of

the triangle , the equation

x 2 2M1 ﬂ2 V 3 2 2 2 2 22 0 0 v x/ (0 1 ) (1 0 o 0 0

represents one of the exscribed circles . 2 From the absolute term of the equation ( 1) in 0 we get " for the radius 7 of the inscribed circle 32 CIRCUMSCRIBED TRIANGLES .

’ s r exscribed and for the radiu of an circle ,

z z z a x/ (ul ﬂuz (9) d b

From these expressions for the radu we deduce if s be

- of the semi perimeter the triangle ,

2 2 z 2 2 J EN? 66 ) (NZ d ) (M3 ( 2 ) d b

B y means of elliptic functions the equation of the circle inscribed in a circumscribed triangle can be written in a form similar to that of the circumscribing circle of an inscribed P ’ ’ 6 . x a sin cos ( for on S triangle utting y p a point , ’ ’ of az the equation the tangent at , y is

a: cos ¢ — 1 = O ;

and if we suppose thi s tangent to pass through the point

m= OOS y sin ¢ , HL, on the confocal ellipse we have

sin c sin 11 p cos ¢ cos 30 1 . ( )

Comparing this equation (11) with

g 2 cos ( ) o sin 5 sin t k 8111 0 cos a 1 c s it 9 g ) ,

M is: c S a' , inz

34 CIRC UMSCRIBED TRIANGLES .

45 S b of . how , y comparing the equations the circles in 4 of of E x . 4 E x . 1 and , that the points contact tangents to a S of t conic , which are parallel to its chords intersec ion with a circle S lie on confocal conics passing through the points of ’ of S intersection of the common tangents S and . f of 46 . I two vertices a triangle circumscribed about an

f ae a of ellipse move along con ocal hyperbol , show th t the locus the centre of the inscribed circle is a concentric ellipse .

v f I f two vertices mo e along con ocal ellipses , show that the centres of each of the inscribed circles lie on concentric ellipses . A s f 47 . circle touche the tangents drawn rom a variable

on S f point a conic to a con ocal conic , and cuts orthogonally ’ a circle S ; show that its centre lies on a conic passing through ’ the intersection of S and S . 8 G n b 4 . iven a tria gle circumscri ed about a conic and a f on o . point the curve , to ﬁnd the envelope the director circle I f the conic referred to the triangle be written in the — - — icc ib ib O of form M J B J y , the equation the director

l mca S m 1 Comic A t c O r . circle is 1 g n ( ) ( s ,

S S S on where 1 , 2 , , are the three circles described the sides B ut of the triangle as diameters . the envelope of the — ’ la Mic 7 O circle subject to the condition J MW 3 , , ' b S S c'w S S is ( 6a 8 2 8 3 B g l / I 2 O which represents a bicircular quartic passing through the vertices and the feet of

a see the perpendiculars . We can lso that the envelope is a bicircular quartic , from the fact that it cuts orthogonally the

n x mics o Co Art. polar circle , and has its centre the ﬁ ed conic ( , 2 293 Ex . , )

W (65 0 7 d a )} { w ay a.“be) ;

’ () C { ey (aa B W } O , CIRCUMSCRIBED TRIANGLES . 35

When the quartic breaks up into tw o circles we see from (2) that these circles must be the circumscribing and nine f x f . I point circles o the triangle the ﬁ ed point is at inﬁnity,

of of t the quartic can break up into one the sides the riangle , and the circle described on the Opposite vertex and the in

ter section of the perpendiculars as diameter . To ﬁnd the point through which the curve p asses in the

of w e x s x of ﬁrst these cases , e pre s that the radical a is the

mcs 883 Co i Ar t. director circle and the circumscribing circle ( , )

la cot A M B cot B ny cot ( 7 : O

touches the circumscribing circle , when we obtain

w hich is evidently the condition that the conic should pass through the centre of the circumscribing circle ; hence if a

so triangle be circumscribed about a conic , as to have the

of centre the circumscribing circle on the curve , the circum scribing and nine -point circles will both touch the director

b of circle . We can verify this result ymeans the invariants ; 3 x2 7 for 1f i me 1 O cl r cumscri b we wr te the co 7 , , and the w g ’ 2 ’ 2 x x s S 0 ing circle ( ) (y y ) , we have

?“ 2 z 2 (m y a 6 2 b g/

3 r 0 x d ~ which , combined with ) / ( the con i S tion that the director circle should touch , gives

93 ‘ r: l 0 e f e & c . Z ; th re or ,

I f a tangent to the curve touch the director circle we can easily see that it must be perpendicular to an asymptote ;

if so hence , a triangle be circumscribed about an hyperbola ,

I) 2 36 CIRCUMSCRIBED TRIANGLES .

one of s that the sides is perpendicular to an a ymptote , the 2 director circle will touch that side , and also from ( ) the circle described on the opposite vertex and the intersection

of perpendiculars as diameter . In all the cases in which the en v elope breaks up into

of 8 curves lower degree, the conic ( ) has double contact with ’ ’ f a A B the polar circle ; in act, putting cos , 6 cos , l 7 cos C in it may be written in the form

2 2 2 a sin 2A (3 sin 2B 7 sin 2 C

— 2 = 2 sin A sin B sin C (a cos A + B cos B + y cos C) O ;

’ ’ ’ a B A 1 for and putting cos , B cos , y the point at to 8 inﬁnity on the perpendicular y, ( ) becomes

2 2 2 i 2 a: sin 2A 6 sin 2B 7 s n 2 C 2 sin A sin B sin 0 7 O .

9 G a 4 . iven triangle circumscribed about a conic , and

x x that a directri passes through a ﬁ ed point , to ﬁnd the locus

of the corresponding focus . f f f x A of I p, be the distance of the ocus rom a verte the

( a f b triangle , p, the ngle subtended at the ocus y the side

A a f A on x opposite , the perpendicular rom the directri , and

e Combs Ar t. 191 the eccentricity of the curve , we have , )

P1 8 1 8 a P1 cos ¢ 1 1 mpg

f s where S , is the square of the tangent drawn rom the focu to

on the A the circle described side opposite as diameter , and pg, p3 f of . Now are the distances the focus rom the other vertices ,

a if , B , 7 are the perpendiculars from the vertices on the

x e la 6 b 0 Z m n directri , we must hav m y , where , , are the areal cc - ordinates of the ﬁxed point ; hence the focus lies

2 2 I [ 8 7710 ”1 0 18 O 2 P1 1 , n 1 3 5 , ( ) CIRCUMSCRIBE D TRIANGLES . 37 which represents a bicircular quartic circumscribing the tri angle .

I f the x of ﬁ ed point is the intersection the perpendiculars , the quartic (2) must be divisible by the circumscribing circle ; for e a parabola always satisﬁ s this condition , and then the f fo cus lies on the circumscribing circle . The other actor is the polar circle of the triangle ; for this circle is orthogonal to the director circle , and it can easily be proved that a circle orthogonal to the director circle , and having its centre on a

x e d f . directri , passes through the corr spon ing ocus

I f to the directrix is parallel a given line , the locus is a — c ma f s L x 2; circular ubic , as y be also proved as ollow et , y, be the perpendiculars from the focus on the sides of the

a triangle ; , Q, 7 the angles between the sides and the given line then we have

2 2 2 2 x + 2 0x C a 6 ~= O m m + c cOS 6 O os , y e e y ;

b c and , eliminating and from these equations , we get

2 2 2 2 2 a: C S a 2 cos 2 $ z COS ' x c O O (y ) y B ( ) y ( f ) , which represents a circular cubic passing through the vertices

the and the centres of the circles touching sides . i 50 . A triangle s circumscribed about a conic so that the

f x of lines, which are drawn rom each verte to the point the

to on opposite side with the circle escribed that side , intersect the curve show that the circle inscribed in the triangle formed by the middle points of the sides passes through the centre of the conic . F F ’ 51 . A ar e conic whose foci , is inscribed in a triangle ; if F on e of w lie the polar circl the triangle , sho that an equi ’ e F for lateral hyp rbola can be described , having centre , and ’ passing through the feet of the perpendiculars from F on the sides . D T G 88 CIRCUMSCRIBE RIAN LES .

I F on of f lie one the circles touching the sides, show ’ n F for f that a parabola can be described havi g ocus, and ’ passing through the feet of the perpendiculars from F on the sides .

52 . Show that the intersection of the perpendiculars of a triangle formed by three tangents to an equilateral hyperbola and the centre of the circle passing through the points of contact of the tangents are conjugate with respect to the curve .

m — 5 : 8 . A i s t 1 0 triangle circumscribed about the conic g , so that the radical axis of the circumscribing and polar circles passes through the centre of the curve show that the centre of the circumscribing circle lies on the conic

2 z g 4 4 2 2 a x b ( 5 6 ( 4 6 y . SE F- CO JU A E TR A ES III . L N G I NGL T .

5 of of 4 . To ﬁnd the equation the polar circle a triangle

- self conjugate w ith regard to a conic .

E x 1 W e have seen in . that when a conic and circle are

2 2 90 y f S written in the orms 2 2 a 6

’ S E (w (9 o the discriminant of S MS is giv en by the equation

7 + + 2 z 2 ail — 13 b — la 11

’ But if S and S ar e referred to their common self-conjugate triangle , we have

3‘ 2 6 C7 9 + , B r

2 2 2 a sin 2 A 6 sin 2B 7 sin Q C

2 sin A sin B sin C

” ’ w e a are cc - of t of S S h re , y the ordinates the cen re being taken so that the result of substituting these values is nega ' and S so efﬁ e of J tive unity, that the co ci nt g/ is unity.

’ B ut when S and S ar e w r itten in the forms (2) and (8) 2 ’ the discriminant of S 11 S is

’ ’ t “ ’ 213 a cos A ) QR B cos B) (la ZR y cos C)

w h er e R is th e r adius of the cir cumscribing circle . Thi s F - J T T 40 SEL CON UGA E RIANGLES .

2 2 z 11 12 lz equation must coincide with hence , if 2 , g are the roots of we have

2 ? 3 ’ ’ ’ 12 i 819 a A M 2 n 6 y cos cos B cos C

12 h h br Ex a s . 1 but 1 z a , as we have een in , and

2 4 B cos A cos B cos C;

2 2 l 6 therefore f 4 a B 7

- A if x & c . cc of o gain , l , y,, , are the ordinates the vertices f the

$ 3 ’ ’ 3 w e cos A triangle , have ﬁ y 22 therefore

’ ’ ’ ’ 2 2 h 2 R a A 213a 6 $ 50 , cos B y a 2 3 my, from (4) since

l z s fj y 2 z 2 1 a M e w s la 19 2 2 , and z g, , a b

Ex . 1 hence, since as in ,

; ; a h cf h a fa ( f ) ( e( r) 1. we obtain

0 and, similarly 64

’ 2 2 r 6 J: y a + 112

2 6 50 33 x x 6 1 2 z g 5133 501) 61

’ ﬁ of S Thus , nally, the equation can be written

2 2 20 ' x zz 2 x3 x e 50 Jﬁ x x x t l g/ l g/ z ysg/ ( l z s g l ) a

- 42 SELF CONJUGATE TRIANGLES .

57 . Show that the circle circumscribing an equilateral triangle self-conjugate with regard to the conic

has double contact with the bicircular quartic

2 2 2 2 2 2 2 2 — 4 a b ( ) z 2 ‘3 a _ 8 5 8 a

58 of of = . To ﬁnd the locus the vertices equilateral tri f- angles sel conjugate with regard to a conic .

L + — O c S a + sin a = O et the conic be , and let y , ;2 %2 x cos B + ysin B = O be lines through the origin parallel to

f t x ( the two sides o the triangle which meet in he verte my.

x f Then , e pressing that these sides orm a harmonic pencil

n v f x with the ta gents to the cur e rom , y, we get

2 — OOS a COS B(x b )

2 2 z 2 2 — — — - — (x + yJ a b ) cos (a B) ( x y 0 ) cos ( Ll

2x sin y (a B) O . ( l )

— 1 - a anc z 2 w cos ( B) B ,

where COS w Sill w

hence ( 1) becomes

9 2 3 2 { P r J 2 y ( 23 5 )

which represent s a q uartic cur v e w ith a node at the origin . - SELF CONJUGATE TRIANGLES . 43

2 2 I f at 86 O , the locus breaks up into two imaginary

conics . z For the parabola y 47a O the locus is

fb (8x 7m) 47m(Jr 8m) O .

- 59 . Given a triangle self conjugate with regard to a

if x x conic , a directri passes through a ﬁ ed point , to ﬁnd the

s of f locu the corresponding ocus .

I f £0 f a conic be re erred to two lines , y, at right angles to e x ach other through a focus , and y the corresponding directri , ? 2 2 of is J: b 8 if tw o the equation the curve y 7 ; hence ,

w e m 1 2 points are conjugate , have m 9 1 2 which g ma S e w 3 er endi y be written 3, ﬁy, here ( , y are the p p culars f the x S e rom the points on directri , and 3 is the squar of the tangent drawn from the focus to the circle described I f on . the line joining the points as diameter , then , we are

f- x given a sel conjugate triangle , and a directri pass through

x b Za 772 M O the ﬁ ed point determined y the equation 6 y , the corresponding focus will lie on the bicircular quartic

‘ ‘ 2 ZS g S g i W S 3 8 1 O .

This quartic coincides with the envelope of the director

48 w E x . circle in , and ill break up into circles in the same

cases . n f - n 60 . Give a sel co jugate triangle with regard to a

: i U conic f a directrix touch a conic inscribed in the triangle , sho w that the corresponding focus lies on the director circle

of U.

6 A f- w e 1 . triangle is sel conjugate ith r gard to a conic ; show that the feet of its perpendiculars form a triangle cir cumscribed about a con focal conic . - 44 SELF CONJUGATE TRIANGLES .

6 - 2 . Given a self conjugate tr1angle with regard to a f x m conic , and that hal the length of the a is ajor is equal to

of x the radius the polar circle , to ﬁnd the envelope of the a is minor . I f two points are conj ugate with respect to the conic z; 3: we have

2 $l yl yz 6

2 z ma i S 6 e 1 which y be wr tten 3 ﬁy, ( ) where S , is the square of the tangent drawn from the centre to the circle described on the line joining the points as d f iameter , and B , 7 are the perpendiculars rom the points if A o x . Now B C A n the a is minor , , , are the angles and

a i f the are of the tr angle , we can verify the ollowing identical relation

2 S tan A + S tan B S C 2 A 15 tan A tau B 0 2 1 2 , tan tan , ( ) where t is the tangent drawn from a point to the circum 2 2 2 b of B 15 a 6 scri ing circle the triangle . ut , since the circumscribing circle cuts the director circle orthogonally ; ? and r 2A cot A cot B C r cot , where is the radius of the

f for polar circle ; hence , rom and similar relations the 2 other sides , ( ) gives

(37 tan A ya tan B aﬁ'i

r and the envelope , putting

tan A a tan B a tan 0 : O By y ﬁ , - SELF CONJUGATE TRIANGLES . 45 w hich represents a conic inscribed in the triangle and concen tric with the circumscribing circle . 6 G - 3 . iven a self conjugate triangle with regard to an

- of ellipse, and that the latus rectum is equal to the radius

w x the polar circle , sho that the major a is touches a conic confocal with the conic (8) in the preceding example .

64 A e S of - con u . circl touches the sides a triangle self j gate with regard to a conic U ; show that the centre of S lies on the equilateral hyperbola having double contact with U at a pair of points which lie on a tangent to S .

A of f-c n u 65 . circle S touches the sides a triangle sel o j 2 gate with regard to a conic U 1 = 0 ; if the tan gents drawn from the centre of S to U touch at the points P t s f P , Q , show tha the tangent rom , Q to the confocal conic x y 1

S . ‘ 4 ? all touch a 6 a + 6

66 I f o of i - . circles t uching the ides tr angles self conjugate 2 s 9 with regard to the conic U + 1 : 0 have their centres 1) on x a fi ed line, show that they have double contact with a fixed conic whose foci are the points where the fixed line meets U. And if U , conversely, a circle have double contact with , the chord of contact being perpendicular to the transverse x - a is, show that it is inscribed in triangles self conjugate with regard to the conic

2 2 2 ﬁ 2 6 ac 0 ( 2 O ( ) g/ fy ,

where f is an arbitrary constant .

6 A 1s 1nscr1bed - 7 . circle in a triangle self conjugate with 2 y to c U + —— O c t regard the onic , and has its en re on b2 — 46 S ELF CONJUGATE TRIANGLES . the director circle of U ; show that it touches the conic

2 ? x y 1

° 4 ? 2 ( 44 6 a 6

68 I f t s f-con u . two circles ouch the ides of triangles sel j

e U s gate with r gard to a conic , how that their centres of similitude are conjugate with respect to U. 6 A f- 9 . e e circle inscribed in a triangl , s l conjugate with regard to a hyperbola , cuts the hyperbola orthogonally at a point P ; show that P must be the point of contact of a tangent perpendicular to an asymptote . I V TR IANGLES FORMED B Y TW O TANGENTS AND RD OF O A THEI R CHO C NT CT .

0 of 7 . To ﬁnd the equation the circle circumscribing the

’ f b x triangle ormed y the tangents drawn from the point , y

0 n

1 of O . to , and their chord contact the conic 3 z,

The equation of the circle must be of the form

1 ( lac my + 12)

Substituting in this equation the cc -ordinates of the point

’ ' ’ x n M m + 1 , y , we get ( y and the conditions that ( ) should represent a circle give

’ 1 M 1 my 62 Hence the equation (1) becomes

’z 2 m + y + 0 ) as + y - 0 ( 62 ( ) 48 TRIANGLES FORM ED BY TW O TANGENT S

A R of lso , if be the radius the circle , we have

2 (55 + y (s y

7 ? 2 2 2 ? 1 3— + ﬂash y ) f20 (93 y ) 24 27

hence 2

? 2 a b

’ ’ ’ I f x f . 0 where p , p are the distances of , y from the oci is

the angle under which this circle cuts the director circle , we have

2 2 M61 6 23 cos

o 3 therefore , fr m ( )

2 2 2 ,2 z 2 (a; b ) p p cos ﬂ {HUM a y

2 2 2 6 6 2 2 2 2 ( or ) sin 0 y 6 ) x {1/ ( c 22 1 62

’ ’ H m or s ence , if , y lie on the director circle the inver e of the director circle with respect to the circle described on the n i li e joining the foci as diameter, the var able circle touches

the director circle . 1 S o 2 x 7 . ince the equati n ( ) in the preceding e ample is 2 l 2 C a: C i/ I I u l s x r es ec na tered, if we sub titute J, ” 2 for , y p a, x tiv el b y, it follows that two points connected y the reciprocal 2 a m 0 r m 0 O relations n y, yl z , a , my, , and the points

50 TRIANGLES FORMED BY TWO TANGENTS

I f J f P on passes through the oci , show that lies a circle

r of on passing th ough the foci , and the centre the circle a x b ﬁ ed conic, both of the loci, in this case , being divisible y the axis major . 4 7 . The circle passing through a point P and the points of contact of the tangents from P to the conic

V 1 O has its centre on the director circle of V ; show that the locus of P is the bicircular quartic

2 2 2 2 z 2 z x e 2x 857 6 Z 86 o O ( f f ( ) y ( ( .

- f 75 . To ﬁnd the co ordinates of the centroid o the ’ ’ f b w angle ormed y the tangents from , y to the conic

2 2 x 9

2 2 ( 6 6 and their chord of contact .

L a cc- i f et , B be the ordinates of the m ddle point o the

of o chord c ntact , then we have

but it can be easily seen that

/ x

hence we have x 1

’ - y 1 + AND THEIR CHORD OF CONTACT . 51

I of i on the u f the centroid the tr angle lies c rve , we ﬁnd

2 Z m y 4 O 5 2 (3 6

' ’ u az v b u for the loc s of , y, after di iding y the eq ation of the curve .

6 cc- i of of 7 . To ﬁnd the ord nates the intersection the perpendiculars of the triangle formed by the tangents from 2 2 ' 1 O x . , g/ to the conic and their chord of contact ; Z,

L x cc - of 50 et 1, y1 be the ordinates the centroid, and 2, y,

of i those of the centre the circumscrib ng circle , then we b have , y a known geometrical relation ,

(13 3332 Cl]

- x cc of . where , y are the ordinates the point we seek ’ ’ w Ex 70 B x x d . ut 2, yz are e presse in terms of , y in 5 w Ex . 7 and ] , yl in hence we obtain

2 ’2 9 2 y

ﬂ 2 a + b

2 a? 6

’ ’ z e S f 1 ww o b ince rom the equations ( ) we have yy , it follows that these two points are conjugate with respect to the director circle . 2 2 a W e hav e also ? x 7 ./

’ if 56 w hence we are given , y, we determine , 3/ as the inter E 2 52 TRIANGLES FORMED BY TWO TANGENTS

of of x section the polar , y with regard to the director circle and the equilateral hyperbola which passes through the feet

of x Combs A 81 rt. 1 the normals to the curve from , y ( , , E x . f 77. Tangents are drawn rom a point of the curve

2 \ 2 2 $ ? 2 ? (a 5 ) a ( “4 g.

2 2 w _ y n + 1 = O ° to the co ic 2 2 a b show that they form with their cord of contact a triangle whose intersection of perpendiculars lies on the director circle f o U.

78 . To ﬁnd the equation of the polar circle of the tri angle formed by two tangents to a conic and their chord of contact .

cc- n 6 The ordinates of the centre are give at Ex . 7 ’ w e b x x z and ﬁnd the absolute term y e pressing that , / is the m cf 1 O pole ? with regard to the circle ; hence we a ﬁnd for the equation sought

z 2 z z z (m + y ) — a + o2 + a + o a

2 c z z ’z 2 + a + o + cc - = 0 + . 1 Z ( )

I r of f is the radius the circle, we ﬁnd then

2 ’2 a bz (w y at2 62) AND THEIR CHORD or CONTACT . 53

The equation (1) can also be written in the form

2 ’2 2 2 2 2 ( 7 + 6 + (x a + 6 b2

2 2 2 — a — + — 1 = 0 2 %E ,

which gives the equation of a pair of chords of intersection f n I f o a d . s the circle conic the conic is an ellipse , these chord

never meet the curve in real points .

79 . To ﬁnd the invariant relation connecting a conic and the polar circle of the triangle formed by two tangents to the

curve and their chord of contact . I f w e make the equation (1) in the preceding example identical with

2 2 — x + y 2ax

2 ’2 2 / 2 c 2 z x 41 + b + 2 2 g Q 2 2 z z 16 b k — a - 0 2 x y + “2 62

2 2 therefore

? 2 a 6

2 z a + b

2 z z 16 a b

2 2 hence ( 0 6 . 54 TRIANGLES FORMED BY TWO TANGENTS

We can ﬁnd this relation also as follows — I f two conics z ? 2 2 u ax b 02 0 z cr 0 can be red ced to the forms y , y , ’ ’ n b 6 9 AA O they must be con ected ythe invariant relation , 3 2 ’ or two roots of the equation 4310 9 76 6 79 A 0 must h n . Ex be equal with opposite sig s pressing, then , t at two b roots of the equation at Ex . 1 (10) are connected y the rela 2 h 12 0 if tion f , we obtain the relation hence the

1 x J circle ( ) cut orthogonally a ﬁ ed circle , the locus of its f u of centre is in general a cubic . I J passes thro gh a pair

of u for vertices the curve , the loc s reduces to a conic ; putting

2 2 z 2 2 x Q a or It 2 a y Bg/ , 63/ , f b we get from a ter dividing y y, 62 2 2 2 2 9 6 26g) . 2B

I f J w u the circle is concentric ith the curve , the loc s is the conic obtained by considering it as a constant in the

u h ma u eq ation T is conic, it y be observed , to ches the tangents of the given curve which are perpendicul ar to its 4 4 2 2 ( 6 4 6 66 6 2 m if 10 w w asy ptotes , and ? 2 ill coincide ith the a 6 given curve . f 80 . Tangents rom a point P to the conic

form with their chord of contact a triangle whose intersection of perpendiculars lies on U show that P lies on the inverse of U with regard to its director circle . S m h how also , in the sa e case , t at the polar circle has double contact with the bicircular quartic

b g/ AND THEIR CHORD OF CONTACT . 55

8 f he of l b 1 . I t polar circle the triang e formed y the

f x U 1 tangents rom , y to the conic 2, a; , and their

of at an 9 w chord contact , cut the director circle angle , sho that

2 2 2 2 x + 9 _ a _ 6

2 2 2 2 4 a b (66 6 )

82 A P x x . point moves along a ﬁ ed parallel to an a is

of 1r cle f of. the conic ; to ﬁnd the envelope the polar c o the triangle formed by the tangents to the curve from P and their chord of contact . ’

P x a Ex . 78 utting a constant in the equation , we see that the circle cuts orthogonally the ﬁxed circle

? a] z co + / — x — b = 0 g , a

’ and cc - d of c , eliminating y between the or inates the entre , we ﬁnd that the centre lies on the ﬁxed conic

4 2 2 a a b 2 2 z 2 4 — 4 2 a x + b y a b 0 a

hence the envelope is a bicircular quartic . 2 ( I f a for t x 1 we put 2 the direc ri , the circle ( ) and the conic (2) have double contact with each other at tw o points on x u tw o the directri , and the envelope then breaks p into circles . These two circles are imaginary for the ellipse . ? ? — 2 2 I f a b 0 a O of ) , the equation the circle becomes

x + y + 56 TRIANGLES FORMED BY TWO TANGENTS which is altogether ﬁxed hence we infer that the tangents to the curve from any point of either of the lines

i ew = 0 form with their chord of contact a triangle whose intersection of perpendiculars is the ﬁxed point

’ = y 0 .

The rectangle under the segments of the perpendiculars is 4 hi also given in t s case, being equal to ? a

8 is The circle ( ) has double contact with the curve, and always imaginary.

of for The same property holds , course , also the lines

2 2 2 6 (a 6 ) cz yz 0 °

and the equation of the corresponding circle is, then ,

(4)

These lines and the corresponding circles are real for a hyperbola whose director circle is imaginary. It may be observed that the curve is its own reciprocal with respect to one of the circles (8) and for the polar of ’ ’ an w 3 y point , y with regard to the circle ( ) is

’ ' z ’ Z wx b w + x ( Z O yy )( ) ,

the and this line, subject to condition

2 ’2 y 2 2 a 6

is v . a tangent to the cur e This is , of course , also evident

58 TRIANGLES FORMED B Y TWO TANGENTS

i B 1 0 n ut ; 2 , bei g the tangent to a confocal f a 0 ’ thr ou h x e is of conic g / a bisector the vertical angle, and mw and 1 0 this line 2 must be equally inclined to a the axis thus we have

or the locus is the confocal conic

2 z 2 x d 6

° 2 2 2 a 6 dz 6

2 S u n a O b a ince the eq atio B 7 is satisﬁed y B y, we see that the centre of the circle exscribed to the base is

on . also , in this case . situated the curve 2 ’ ' ( 0 37 b g ° The equation 0 0 represents the equila 93 i teral hyperbola which passes through the feet of the normals

’ fr om x to 1 O of g/ the curve , and if the polar ? a ’ ’ x x y cut this hyperbola orthogonally at , y, we must have

’2 2 '2 2 m y y x 0 .

’ ’ ’ ’ f x w 0 x Taking the actor y y , we ﬁnd that , y must satisfy the equation

2 2 2 66 b 66 b w s w 1 hich coincide ith the conic ( ) hence we see that , in the above case , the base is normal to the equilateral hyperbola which passes through the feet of the normals to the curve from the vertex of the triangle .

86 . In the preceding example show that the product of the diameters of the curve parallel to the tangents is equal to the square of the diameter parallel to the chord of contact . D F AN THEIR CHORD O CONTACT . 59

F ’ 87. x z rom a point , / tangents are drawn to a conic , to ﬁnd the cc -ordinates of the focus of the parabola having o double contact with the curve at their points of c ntact .

L + l 0 of et 2 be the equation the conic then z 22

— = v ) 0 is the tangential equation of a conic having double contact with the curve at the points of contact of the tangents from ’ ’ 2 x but fﬁ of v , y ; for a parabola the coe cient must vanish ,

h it 1 w ence , and

2 2 2 (x + 60 ) A

l represents the required parabo a ; hence , to determine the f W e Combs Ar t 279 E x oci , have ( , . , . )

' ’ ’z ’2 ’ ’ 2 — 2x x + x + c O x + x O g y y , y y ;

,2 ,2 x x I ( y l ( y 2x 2 = therefore ’ y y ’ ” 2 y x

From these expressions it follows that if the vertex of

n x to f the tria gle be ﬁ ed , and the conics belong a con ocal

of lae x . system , the foci the parabo will remain ﬁ ed ’ ’ I f x on u of , y lies a line thro gh the centre a concentric

93 on f . circle , , y lies a con ocal conic ’ G x x as —L confo iven , y, we determine , g/ follows et a

b u 90 cal hyperbola be descri ed thro gh , y, then the tangent to

b x the hyper ola at , y intersects its asymptotes in the cor ’ '

of w . responding positions , y

88 F e . rom a point / tangents are drawn to a conic ;

of prove that the centre the equilateral hyperbola , having

b n the v t o n of dou le co tact with cur e at heir p i ts contact , is

’ the inverse point of x y/ with regard to the director circle . ( 60 3

V — H THE . CIRCLES HAVING DOUB LE CONTACT W IT

CURVE .

8 con 9 . There are tw o systems of circles having double

a the of tact with conic , chords of contact each system being

one parallel to of the axes of the curve . z i y — ’ I f f 1 O of x m= 0 2 2 is the equation the curve , and an 6

of of of n that a chord contact , the equation the correspondi g circle is 2 2 z ’ 2 ” 2 x g/ Qe x w 6 513 6 O ; (1)

if ’ O o and y y is the chord of contact , the corresp nding circle is / 2 g 2 2 2 2 x y 20 0 ? ( ) t)

I f a is the abscissa of the centre of the circle and r its radius, we ﬁnd (3)

if 3 of of 7° 1ts and { is the ordinate the centre and radius , 2 B2 2 0

We may convenientlysatisfythe equation (3) by assuming

= = a c s 0 r . co , b sin 0

From the equation (4) we see that the radius of a circle of the system (2) is in a constant ratio to the distance of its centre from either of the foci . CI CLES H O T T W TH THE R AVING D UBLE CON AC I CURVE . 61

90 To f cc - di . ﬁnd the di ferential equation in elliptic or nates of the system of circles having double contact with a

conic . If we write the equation of the circle (1) in the preceding example in the form

2 2 2 2 2 2 x 0 0 cos 9 6 9 0 y cos sin ,

and transform this equation to elliptic cc -ordinates byassuming

2 2 2 z — 2 w + = + v 0 cm= u v y p , , ,

2 2 2 2 9 C sin 0 O we get M v cos , which is equivalent to the relation

i “ 1 f cos 1 a

b ff n hence , y di erentiation , we obtai

dre 2 — ( ) v )

From this equation we see at once that the tw o circles of the same system which pass through a point are equally l inclined to the confoca s through the point . In a similar manner we ﬁnd that

“cl/r v dv i O 3 2 “ 2 2 — 2 2 — 2 z 2 ( l my 4 ) (M O H mm v >

of is the differential equation the second system of circles .

9 T o tw o i o 1 . ﬁnd the angle between c rcles f the same system which pass through a point .

I f 9 r di of d s 3 , z are the ra i the circles, and the di tance n between their ce tres, we have

2 tan g (p DOUBLE 62 CIRCLES HAVING CONTACT WITH THE CURVE .

where qb is the angle sought . But

7 6 sin 9 44 1) 0 d c cos 0 cos 9 3 1, 2 sin 2 , ( 1 2) and from (1) in the preceding example we have

= - u a cos v a cos 33 691 (1)

1 ( (2 hence tan 5 p iti t; JO} )

We could ﬁnd this expression at once from the differential equation for when two curves are represented by the diffe r ential Pd j : 0 < ; equations u d , the angle 1 between them is given by Q A ; tan 9 4 2 P M 0

E x 90 2 O whence from . ( ) we have the result already btained . From (2) we get 2 2 2 2 2 2 2 6 97 (66 0 ) g/ 6 0 cos 2 b pp

sin (15 2C

’ of x f f where p, p are the distances , g/ rom the oci ; hence , tw o being given the angle between circles of the system , the locus of their intersection is a curve of the fourth order of I f which the foci are nodes . the angle is right , the locus is a concentric conic passing through the foci . In a similar manner for the other system of circles we ﬁnd

z 2 V 63 1 w/ (M ) ( ) tan 9 I z ’ v ) L CIRC ES HAVING DOUBLE CONTACT WITH THE C URVE . 63

sin ¢ = 2o

where a is the angle between the tangents to the curve from

as m 9 8 . 2 Co bs 6 . 6 1 E x Ar 2 x . Ar t. t E , y ( , , , and , From these values it readily follows that the circles

an P on x described through y point a directri , to have i double contact w th a conic, are at right angles to the tan gents from P.

9 I 2 . I f t is the angle which an external common tangent

of x of two circles the system makes with the a is, we have

94 6 9 sin 9 b 2 (sin 1 2) 1 0 13 0 9 (01 02) 0 cos 01 cos 02 o

2 v ) 9 from Ex . 1 (1)

n x hence, supposing the give curve to be an ellipse , the e ter nal common tangents are parallel to ﬁxed lines when the n f intersection of the circle lies o a con ocal hyperbola . I f n f m 93 . circles having double contact with a conic or

of of n 1 with a single point intersection each a polygon , th o o i n x f whose vertices move al ng confocal con cs , the verte will also move along a confocal conic . I f 0 0 O n 1, 2 n are the parameters of the circles, we 9 see from Ex . 1 ( 1) that we must have

0 0 6 9 & c . 1 2 a constant , 2 3 a constant, ,

0 e h o 0 . whence 1 is giv n , w ich proves the the rem For a H T T W TH TH 64 CIRCLES AVING DOUBLE CON AC I E CURVE .

semiaxes of o o o s triangle the p l , M2, ”3 the three c nf cal c nic are connected by the relation

3 2 2 ? 2 “ a 0 . (In ”2 us ) mm ?»

I a! s 94 . f is the distance between the centres of two circle

f e o the same syst m , we have

1 d 0 0 9 2b 0 0 sin 9 0 (cos 1 cos 2) sin A( 1 2) 3, ( 1 2 )

2b x/ { W 52

2 2 260 x y

From this expression we see that the theorem in the last example is also true if we substitute for confocal conics ” concentric , similar , and similarly situated conics . 95 I f A B of . f , are the base angles the triangle ormed by f of st E 89 the centres o two circles the sy em at x . and a point of intersection , we have ,

‘ ﬁ — T l ‘ t f g d tan 3 A B 2 tan 5 r + r + d o — 2 l 2 u ,u )

Ex 91 from . b 2 tan §A y o w/ (a

‘ — — z tan 5B bv O V W hence we see that either the product or ratio of the tangents

f s A B of of hal the angle and is given , when the intersection f the circles lies on a con ocal conic . of This property is, course , also true for the system of x E x 89 circles which touch the curve e ternally ( . and

h ur de ene since t ese circles become tangents, when the c ve g

see rates into a parabola, we that , if tangents to a parabola

or of intersect on a confocal parabola , the product ratio the

E 66 CIRCLES HAVING D OUBLE CONTACT WITH TH CURVE .

f 99 . If we seek the intersection o the circle

0 2 2 y m + y + 2b ar e z

W I th b an asymptote of the curve , y putting

2 0

- 1 b

and if o 2a p l , pg are the ro ts of this equation , we have p l p2 ; hence a circle having external double contact with a hyper bola cuts off from the asymptotes constant intercepts equal to x f the major a is o the curve . 00 1 . co e The polar of l , yl with regard to the circl

2 2 2 x + y + 2 — g/ y — C £,

2 C 4 wx1 + yy1 + y (0 2 2 1) 52

its envelope , when the circle varies, is the parabola

2 — + 4 = 0 (y g 1) 1 .

P 0 x 0 for f 1 utting y1 , 1 a ocus, ( ) becomes

2 2 46 /a

o o

and the parabola then has its focus and directrix in common with the curve ; hence we see that the feet of the perpen

i ular s f d c from a ocus on its polars , with regard to circles CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 67

x having e ternal double contact with the curve, lie on the corresponding directrix . 9 1 s f E x . 101 . Taking two circle o the system at 8 ( ) namely,

2 2 2 2 2 2 2 2 2 2 2 26 97 612 0 50 6 : 0 x + 29 x 58 3 513 6 0 g/ 1 1 , y 2 2 , we have for their intersection

2 2 2 x y 6 x1 + x2 ;

? x c 2 a ca a 933 therefore y ( i ) ( 1: n) ( t 2) hence we see that the square of the distance from a focus of a point of intersection of two circles having internal double contact with the curv e is equal to the product of the distances

the f from same focus o their points of contact .

0 I f tw o of 1 2 . circles the same system have double con

i a t of is tact w th conic , show tha the product their radii in a constant ratio to the product of the distances of a point of their intersection from the foci . 0 E x 89 8 I f . 1 1 . two circles of the system at ( ) intersect in ’ ’ x i of the points , y , the equation the circle passing through their points of contact is evidently

2 ? ’z 2 E 50 + Jo 25 S y 0 .

I f S x f f is ﬁ ed, and the conics orm a con ocal system , the ' ’

of x z t S . locus , y is a circle cutting orthogonally ’ ' I f th e x i x points , y are ﬁ ed , and the conics form a

a S C confoc l system , has double contact with the artesian oval

2 ? ’2 ” z ' x x 6 c x 0 ( y y 1 x .

’ ’ I f is r of x t is the radius given equal to , the locus , : . y the conic 2 2 2 2 2 “ 2 ( ﬁ g/ 6 (2a 6 ) x a (26 F 2 68 D B LE HE CIRCLES HAVING OU CONTACT WITH T CURVE .

’ ’ S ac i I f the tangents to from , y contain a given angle , ’ ’ w i the polars of , y, with regard to the curve, are touched by a confocal conic . n mm 0 104 . Ta gents parallel to the line y are drawn 89 to the system of circles at E x . show that their points of contact lie on

2 z z 2 z 2 2 (0 m a ) y m xg/ 6 0 0 . Show that this conic passes through the foci and has double contact with the curve . 2 2 2 5 I f S s x 2am 2 k 0 x 10 . y 3g represents a ﬁ ed

x S of Ex . circle , the radical a is of and a circle the system at 89 (1) is

? 2 ” 2 2am 2By la 8 90 6 O and this line is a tangent to

a 2 k? 62 0 1 6g , ( ) which represents one of the parabolas passing through the

of For o points of intersection S and the curve . the sec nd s x ystem of circles the radical a es touch the other parabola, which may be described through the same points .

S is b of When the circle touched y a circle the system, the radical axis is the tangent at the point of contact hence to construct the points where S is touched by four circles of

of S the system , we draw the common tangents and the para bola and then the points of contact of these tangents are the points required .

106 . A S A B C D . circle meets a conic in the points , , , Four circles having internal double contact with the curve

e i S x ar descr bed to touch , and also four circles having e ternal double contact ; Show that the chords of contact of the ﬁrst CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 68 system intersect those of the second in the centres of the sixteen circles which may be described to touch the sides

of AB C B OD & c . see E x . the triangles , , (

107 f of . To ﬁnd the locus o the centres of similitude x a ﬁ ed circle and circles having double contact with a conic . ’ ( u of x 0 The p and equation a circle , whose centre is , and

r radius , is evidently

’ r x 1 p cos ( 0 . ( )

B for f Ex 89 3 ma ut the system o circles at . ( ) we y take

c cos O r : b sin O , ;

if a the s o also , , £3 are perpendicular fr m the foci on the line

9 w 1 , , we have

C OOS w ( cv hence (1) becomes

a - 9 + cot - O- 2b = 0 tan § B § ,

2 t e of i a b 0 h s to . envelope which ﬁ , as it ought be I f w e now write the tangential equation of the ﬁx ed ' ? i 2 E k O c rcle in the form , the equation

26 a - o -F COt —— Oi ' I tan é B L z

’ will represent one Of the centres of similitude Of 2 and the circle The envelope of (3) with regard to 0 will then

iz v . give the locus , ,

2required — — 2 a 0 a b 4 0 4 s , or s (r ) < ) 22 7 w hich represents a conic passing through the foci and touch ’ n 2 ing the common ta gents of and the given curve . 70 CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE

f - If the equation o 2 . in Cartesian co ordinates

(s — d P+ O — yT the locus is found to be

2 116 ’ ’ y +- (x y J (y — 9 V= 0 y ¢ O

We see then that the locus will become a circle if

2 k y 2 2 6 0 in which case 2 has double contact with the confocal conic

Z y + r sR - w

The locus (4) evidently passes through the four points on W here it is touched by circles of the system f 108 . I we take two circles of the system (2) in the pre x ceding e ample , namely,

- — W - = a 0 + cot B QI O a z ﬁ oot 0 26 0 tan }; 1 B g z , tan g g B é 2 r , we ﬁnd that their common tangents satisfy the equations

2 9 w tan 7} ( 1 92) a n awﬁ m

9 if Ex . 1 see hence , from we that , the common tangents are parallel to a given line , or touch a concentric , similar and similarly situated conic, the intersection of the circles will lie on a ﬁxed confocal conic . 9 P 10 . x Through a point , e ternal to a conic , two real circles are described to hav e double contact with the curve ; ’ if a is half the axis major of the confocal conic touching the

of ) is common tangents the circles , and q the angle between U CIRCLES HAVING DO BLE CONTACT WITH THE CURVE . 71

f P the tangents rom to the curve, show that sin 5 q) where

x a is half the a is major of the curve . 0 A 11 . right line touches two circles having double con tact with a conic ; to show that its points of contact with them lie on the same concentric , similar and similarly situated conic . — 2 2 2 2 L x cos w + w 0 x cos l9 + — b 0 = 0 et ysin p , ( ) y sin ,

one res ec be the equations of the line and of the circles, p tiv el x cc - of o y and let , y be the ordinates of their point c n tact then

c cos 0 + b sin 0 cos w = b sin 0 sin w x , y ,

p = b sin 9 + c cos d cos w ;

z‘ 2 z 2 2 c cos 9+ c sin Boos ﬂcos w c sin O w } cos 0

2 2 p 0 sin 0)

’ where a is half the axis major of the confocal conic touching

he the line . T conic (3)

i o h therefore, passes through the points where the l ne is t uc ed by both the circles . We can arrive at this resul t by means of the differential F . Ex 90 2 equation of the system of circles rom . ( ) we have for the circles 72 CIRCLES HAVING D OUBLE CONTACT WITH THE CURVE .

Now not diﬁicult , it is to see that

d” “ 2 v ” is the differential equation of all the lines touching the con 2 2 y

1 0 . B ut w focal conic M hen one of the lines

of m m touches one the circles , ust have the sa e value at the

of c m 4 5 point conta t ; hence , fro ( ) and ( ) we have

2 (ﬁ - Mﬁ ) 2 2 2 (M a )(M 0 )

2 2 b v or, dividing y ”

? f 2 2 ? (a m) (c v ) 6 ( a which is equivalent to the relation ’ Putting a c in w e see that the points of contact of the tangents from the foci to the system of circles lie on the conic

s 111 . To ﬁnd the distance 8 between the centre of two circles of the system which touch a given chord of the curve . Let the equation of the chord be

— — O a 3 cos ; a , acos ; ( ( ) ( ﬁ) and that of the circle

2 2 2 2 (x 0 cos B) 9 6 sin 0 0 ;

x n o w e then , e pressi g that the chord t uches the circle , get

2 z 2 ‘ - e os — { cos J, (a B) cos 9 c ﬂ m ﬁ) sin 0 { 1 e cos 7,

CI RCLEs HAVING DOUB LE CONTACT WITH THE CURVE 74 .

m o C co-o m which, transfor ed t artesian rdinates, beco es

2 2 0 31 0 ; but this represents the two chords of the conics which are parallel to the transverse axis . 4 x 1 1 . Through the centre of a circle having e ternal double contact w ith a conic lines are draw n to the foci show that they meet the circle on the tangents to the curve at the f extremities o the transverse axis . 5 i 11 . Tangents are drawn from a focus to a c rcle having external double contact w ith a hyperbola ; show that they contain an angle equal to that between the asymptotes of the curve . 116 m . To ﬁnd the condition that four circles of the syste should be all touched by the same circle . I f we express that the circle

2 2 z 2 (x — c cos 9) + y — b sin 0 = O touches the circle

- u + — — W = O w f @ Bf ,

0 r we Obtain (a cos B ( i 6 sin 0 .

Now if we put

c cos 9 x b 1 sin 9 = , / y, this relation may be replaced by the equations

2 2 x y 2 2 0 b

(x (3) and thus we see that we have to ﬁnd the condition that four points on the conic (2) should lie on the circle (8) but this is known to be

9 0 9 0 r 2771 1 92 3 , , o 713 CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 75

n s s o f a When this co dition is ati ﬁed, a c n oc l conic passing through the intersection of one pair of the circles will also pass through the intersection of the remaining pair (see E x . 91 (1) If four circles having double contact w ith a parabola are

b m w e b i sum all touched ythe sa e circle , ﬁnd that the alge ra c of their radii is equal to z ero . 1 in 1 7 . Let us suppose three of the circles the preceding

x m to see m 4 if on e a ple coincide , then we fro ( ) that , the os lating circles at the points it 0 on the curve are touched by a circle of the system w hose parameter is w e must have

' 39 + 9 z 0 or 2 , 77 ; hence three pairs of osculating cir cles can be described to touch a given circle of the system; and if circles of the system be described at the points of contact of these oscu

w lating circles , their intersection ill lie on the confocal conic

E x ? , f 2 (see . a a 40

We can also show that the confocal conic passing through the points of contact of one of the three latter Circles w ill also pass through the intersection of the r e maining pair ; (2) the algebraic sum of the radii of these circles vanishes ; (8) the centroid of their centres coincides with the centre of the curve . i 118 . To ﬁnd the equations of the circles wh ch touch three given circles of the system.

6 2 see the m From Ex . 11 ( ) and we that proble is the same as that of ﬁnding the equation of the circle passing through three points on the coni c 1 hence from 76 CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE .

Ex . 1 and if the equation of the circle sought is

2 2 w + y — 2Ax — 2B g/ +

9 0 cos $0 1 2) cos 5402 03) cos 5 03

2 2 B = + - a cos % (01 +

2 2 1 C cos 0 0 2, ( 1 m,

0 c s O é ﬂ cos w, 02) cos (92 93) co ( 3

if R of and, is the radius this circle ,

9 « R . sin sin A(0. a.) A(O. a) 5 sin

The equations of the three other pairs of circles will evidently be obtained by changing the sign of one of the

n 9 0 0 . a gles 1, 2, 3

1 w 19 . To ﬁnd the angle bet een a tangent to a conic and

x w a circle having e ternal double contact ith the curve . Let the equation of the tangent be

2 2 2 2 a: 0 0 8 1» sin = a < o 6 sin w y w p ¢ ( cos ) , and that of the circle 9 E x . 8 $2 (y [i f (see

x f m Then , e pressing that the perpendicular ro the centre of “ 7 9 the circle on the tangent is equal to cos , we get

r 0 0 8 9 : m - B siu p ,

B t n 0 n s . u where is the a gle ought , putti g

a s a 0 p in , B cot ¢ ,

( I w e 0 sin w ( 1 cos a ﬁnd , si n qt CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 77 hence ( 1) gives at once

0 a

Taking tw o circles and the same tangent we have

01 02 (PI SI) hence w e see that the tangents to a conic meet tw o ﬁxed x b circles , having e ternal dou le contact with the curve at angles w hose sum or difference is constant and equal to the angle subtended at one of the foci by the centres of the circles .

tw o m Taking tangents and the sa e circle, we have

- ' 01 02 ( 11 hence we see that a variable circle having double contact with a conic meets two ﬁxed tangents to the curve at angles

w sum d ff . hose or i erence is constant This constant, it is easy ff s to see, is equal to half the di erence of the angles ubtended

of by the foci at the points of contact the tangents . 0 x 120 . To express the angle in the preceding e ample in

m of co- x tan ter s the ordinates , g/ of the intersection of the

C gent and ircle . This may be most readily effected by means of the differential equations of the circle and tangent in elliptic

o- c ordi nates . These equations are

2 x/ { Ou COW CO}

x/ { W COW

9 respectively (see Ex . 0 (3) I 78 C RCLEs HAVING DOUBLE CONTACT WITH THE CURVE .

B ut w hen two curves are represented by the differential equations in elliptic co-ordinates

' Pd 0 Pd dv O “d , “Q , the angle 0 between them is given by

’ PQ { COW 0 W tan ’ 2 ’ 2 PP 0 ) Q Q (0

m 1 2 hence , fro ( ) and ( ) we have

x/ { W “OW 2 (Z i m)

f and there ore ,

f m x C o Trans or ing this e pression to artesian co rdinates , we get

a + ex a — efv

’ w of x m 9 here p , p are the distances , g/ fro the foci ; , there w i of fore , is equal to half the angle h ch the points contact of ’ m x the S m the tangents fro , y subtend at one of foci ( al on s

bs A . Com , rt

121 . as / Two circles are described through a point , g to have external double contact W ith a conic to ﬁnd the angle subtended by their centres at a focus . A circle having double contact with the curve being written in the form

26 62 2 2 “ 3 x a O £ 2 ﬂy 2 ( ) , 0 5 CI RCLEs HAVING DO UBLE C ONTACT WITH THE CURVE . 79

for tw o O m we have , the circles f the syste which pass

or through , y,

f a - a o s — as. ( y), I l a

if ( but , p is the angle sought , we have

- o . . c b S _ (B B ) / 2 2 z 2 2 + a + b — x — 0 Bl ﬁ2 y from which we see that (p is equal to the angle between the ’ m Combs Ar t v m x S . tangents drawn to the cur e fro , y ( al on s , 169 E x . , This result might also be readily deduced from the 9 2 E x . relation ( ) 11 .

122 T W O x w . circles having e ternal double contact ith a conic are described to touch a tangent to a confocal conic to show that the angle subtended by their centres at a focus is constant . Expressing that the cir cle

2 2 w + (3/ - B) has double contact with the conic

2 2 x y

2 2 ( 4 6 and touches the line

— x cos w + sin w = 0 y p ,

1 3 sin w r . we obtain , 1 p

E m r w w e li inating bet een these equations, have

2 2 2 2 ? 2 ( L 0 sin to ) sin w ﬁ 0 (a - p ) 80 CI RCLEs HAVING DOUBLE CONTACT WITH THE CURVE .

s w e hence , if B1, B2 are the roots of thi equation, ﬁnd

2 2 0 sin ( 0

6 1 Bz 2 2 2 O 0 sin w

2 0 “ PO

' 3 1 62 2 2 2 (Z 0 sin w

( is B ut if p the angle sought , 2“ 2 2 52) 0 sin ( 0 2 2 — 2 — 2 n2 c + BB2 a p 0 si w from (I ) and 2 2a m ) 3 2 ( ) 261

’ where a is half the axis major of the confocal conic touching

3 ma m m the line . The relation ( ) y be written in the si pler for

cos 5-

— As a particular case w e have z The circles drawn through the foci to touch a variable tangent to a conic out each other under a constant angle . A n 123 . circle having internal double co tact with the conic

2 2 I 0 cuts orthogonally a circle having internal double contact with the confocal conic

show that the locus of their intersection is

2 2 0 c — 2 —_ O. a: 7 y c 62 1 >

8 CIRCLES HAVING DOUBLE CONTACT WITH THE CURV 2 E .

’ The to S w i s tangent at , y , then , 55 —-- — am cos qb i e + Sin

or f tw w w v r o o . hich , hen B aries , touches one othe parabolae 125 L . et — 2 2 — 2 S z (x a ) + y r 0 be a circle having double contact with the conic

2 y 62

’ 2 — 2 — = S E x + (y B) r O a circle having double contact with the confocal conic

2 Z y b’2

Ex 89 w e 3 . ) and , have W 2

z z { z 7. 7 2” COS (15 therefore 2 2 , 2 0

” w 8 where q!) is the angle at hich and S intersect . ’ ( ) o r T o Hence , when 1 is given , the ratio f to has ne other of two const"ant values . Putting 7’ m we get from (1) and (2)

2 2 2 ’2 2 2 2 a n a B C (6 n Now if the line CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 83 is a normal to the conic 2 2 x y Z2 m2

2 2 2 2 Z2 m2 2 we have l a m B ( ) .

’ s S S out We see thu that , when and each other under a o m constant angle, the line joining their centres is n r al to

2 2 y c

2 2 '2 2 2 2 6 n a 6 72 a which represents one of two conics confocal with the given ones .

I x co- of m S f , y are the ordinates of a centre si ilitude of ’ and S ﬁnd m , we , in the sa e case, 2 2 2 b 72 05 Z ’2 z 2 ( ) w d y 0 (72 i f which repr esents one o four concentric conics . ’ 126 nd . If the circles S a S in the preceding example

w Of o touch one another, sho that their point contact lies n the conic

I S S ’ o f and cut orthog nally, show that they intersect on

2 T fo

2 ’2 03 a

2 o t o m f 1 7. T ﬁnd the or hogonal traject ry of the syste o circles having double contact with a conic . 90 ff We have seen in Ex . that the di erential equation of the system having external contact with the curve is

W ill

2 2 2 x/ { W ( 6001 v ) (0 G 2 A ) 84 CIRCLES HAVING DOUBLE CONT CT VITH THE CURVE .

ff of is The di erential equation the trajectory , therefore ,

2 2 d“ - a ( i v a 2 — 2 u c v 0 , u

S O n Taking the lower ign and integrating , we btai

h — lo g w c o mc w u mw e v B ew c v > } 10 e z w

Z Q Q “ — — 8 10 g tx/ (M “Jh / (u ﬂ ! )

— v a constant.

I of to f, then the eccentricity the curve is equal the ratio of

tw o w . For integers , the trajectory ill be algebraic the system of circles w hich have internal contact with the curve the trajectory is transcendental when the curve is an ellipse, and algebraic w hen the curve is a hyperbola w hose eccentri m m w n . city is equal to g here and are integers me w A w 128 . variable circle has double contact ith a conic ; Sho w that the tangents drawn through one of the points of contact to a ﬁxed confocal conic intercept on the circle seg ments of given length .

29 T w o ff m d 1 . circles of di erent syste s have ouble contact w ith a conic ; to sho w that the intersection of their chords of contact is a limiting point of the circles .

Ex 89 ma 1 2 . Taking the equations ( ) and ( ) in , we y w rite 2 2 2 ’ 2 2 2 S E x y 2 6 x x 8 33 6

2 S z x + y

2 2 ’ 2 2 O 8 6 S E 6Z 5 w hence ( ) { ( W (3/ W } ;

which proves the result stated . CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 85

130 i w . Four c rcles having double contact ith a conic are described to cut a circle J orthogonally ; sho w that their chords of Contact form a rectangle inscribed in J so that each pair of opposite vertices are conjugate with respect to the curve . m i 181 . Show that the second li iting point of the c rcles S ’ f m and S in E x . 129 is the foot of the perpendicular ro the “ ’

as on w a v . point , y its polar ith reg rd to the cur e

8 of mm Of tw o 1 2 . To ﬁnd the length a co on tangent

w the cir cles circles having double contact ith a hyperbola, belonging to different systems . I f the circles

iv — a + 0 ( ) 9 , have double contact w ith the conic

2 a?

2 2 C 6

89 w e have from (8) and (4) E x .

therefore 2 62 w here d is the distance betw een the centres Of the circles ;

8 d c 9 hence 2 e 1

B ut t mm , if is length of the co on tangent ,

2 ; t z 03 (7

2 " 8 I r 2T? m I ( ) 2 fro ( ) ; 8 1

e therefor g z 1) f 86 C IRC LE S HAVING DOUBLE CONTACT WITH T CURVE HE . the two signs corresponding to the internal and external

mm em x co on tangents ; hence, if one circle r ains ﬁ ed, while

sum ff o x n the other varies , the or di erence f their e ter al and internal common tangents will remain constant .

133 . Let S be a variable circle having double contact

and S S x s ff with a hyperbola, I , 2 two ﬁ ed circle of a di erent m t syste having double contact with the curve ; then , if l is a mm S S $ S S S co on tangent of and I , and 2 of and 2 , how that t t . , z a constant A 134 . circle S whose centre is P touches the sides of a S S . I triangle inscribed in a hyperbola f circles 1 , 2 be de

so scribed to have double contact with the curve , that their

Of w S is chords contact intersect in P, sho that the radius of

f and S see in a constant ratio to a common tangent O S I 2 ( 3 E x . 10 , ( ) 5 A x 13 . conic has double contact with two ﬁ ed circles ; f to ﬁnd the locus o the foci .

I ff m w e m E x . f the circles belong to di erent syste s, see fro

132 of one of , that the eccentricity the curve has or other

tw o . w constant values Hence , it readily follo s , that the foci lie on one or other of tw o circles concentric with the circle which has external contact w ith the curve .

I x w i f both the circles have e ternal contact th the curve ,

f m E x 89 of f f m we see ro . , that the distances a ocus ro the centres of the circles are to one another as the radii of the f n i . o c rcles The oci, therefore , in this case , lie a circle pass f ing through the intersection o the given circles .

m of cir cles s s The co plete locus , then , consists ﬁve , be ide the line joining the centres of the given circles . 136 I f b . two circles are connected y the relation

2 (17 2 7° CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE . 87

Show that an inﬁnite number of equilateral hyperbolae can m be described to have double contact w ith the .

w tw o x 137 . A hyperbola has double contact ith ﬁ ed circles ; to ﬁnd the envelope of the asymptotes . Suppose ff m A B the circles to belong to di erent syste s , then if , are

and C of A CE their centres , is the centre the curve , is a

is E x . right angle , and because the eccentricity given (see 132 m CA CB , the asy ptotes are inclined to and at

. m constant angles The asy ptotes, therefore , pass through

ﬁxed points on the circle described on AB as diameter . I f m m m the circles belong to the sa e syste , an asy ptote

ff E x T he cuts o equal intercepts on the circles (see . envelope then is easily proved to be a parabola of w hich f the middle point of the centres of the circles is the ocus .

1 tw o x 38 . A w conic has double contact ith ﬁ ed circles, the circles belonging to different systems ; to ﬁnd the en v el ope of the directrices . The equation

2 2 a: + y — 2ax + k (l ) w 0 i m of n here is variable , ev dently represents a syste co ics having double contact w ith the ﬁxed circles

2 2 2 x + y (1 — 8 )(x + 31 )

of m and since the chords contact , na ely,

w oos ﬁ + 9 0 x sin H— / cos ﬁ 0 ysin , g , ah ff are at right angles to e c other , the circles are of di erent

s system . Writing the equation (1) in the form

2 2 2 2 2 a: y 2 01413 la 6 19 20 19 ( 1? cos 9 ysin 0)

c 88 C IRCLES HAVING DOUBLE CONTACT WITH THE CURVE . w e see that w oos 9 + y sin 9 — p 0 will represent a directrix of the curve when the left-hand member of this equation represents the square of the distance from a focus . This condition gives

2 2 — 2 2 — 2 — = e (1 e ) p 2e ap OOS 9 + /c a 0 show ing that the directrix touches a conic of w hich the origin is a focus .

the 139 . To ﬁnd the envelope of the director circles of system of conics in the preceding example . The conic being w ritten in the form ( 1) in the preceding

x m e a ple , the equation of the director circle is

2 2 2 2 2 (1 e ) (x y ) a (1 a sin 9) 2e ay sin 0 cos 0

2 2 k 2 e a 0 Comics Art. ( ( ,

The envelope is, therefore ,

2 ' — 2 2 2 2 2 2 2 — 2 8 x + 2 6 am+ 15 2 8 G e a = 0 )( g/ ) ( ) ( ) } y ) , which represents a Cartesian oval of which the origin is a focus .

140 A w . conic has double contact ith two circles , the circles belonging to the same system to ﬁnd the envelope of the director circle .

I f a m of , B are the perpendiculars fro the centres the

r ci cles on a line , the equation

Qa ( OS O B B — 2 i sin 9 -T - 0 evidently represents tangentially a conic having double con tact w ith the circles

HAVING TA T WITH TH 90 CIRCLES DOUBLE CON C E CURVE .

We ﬁnd thus, ﬁnally,

z 2 — 2 2 2 2 2 2 2 2 2aw 19 1 0 C 2um 79 ax le + a = 0 y ) ( ) ( y ) } ( ) y ,

a o which represents bicircular quartic, f which the centres of

x the ﬁ ed circles are double foci . I f ff m the circles belong to di erent syste s, the locus is easily proved to be the circle described on the line joining m the centres of si ilitude of the circles as diameter . 142 . G on iven three circles with their centres a line ,

C there is, in general , a single conic having double ontact m with the . Since the tangent from any point on the conic to one of the circles is in a constant ratio to the perpendicular on the chord of contact , it can easily be seen that

—— 0 1 Z/ S I e / S z i n / n ( )

S S S of where I , 3 are the squares the tangents to the circles ,

l mn w and , , the distances bet een their centres, represents

f of the conic . This equation (1) w hen cleared o radicals is

the m the second degree , ter s of higher orders vanishing l identica ly. I n m m if 2 2 2 a si ilar anner , 3 are the squares of the

of i w e w intercepts a line on the c rcles , can sho that

n of is the ta gential equation the conic . I f the line joining the centres of the circles is the major

x of 0 a is the conic , we can easily show that the eccentricity ( ) is given by the equation

2 2 I c 173 mr z nr LES H ING BLE NT T W TH HE E CIRC AV D OU CO AC I T CURV . 91

7 7 r adn o s . I s m where 3 , 3, are the f the circle f the a e line the m x s is inor a i of the curve, we ﬁnd

Zmn

2 4 2 2 27 3 7127 2 7273

I o the I m n t is to be bserved that in above , , are taken so that

Z m n 0 .

I n E 97 of the ﬁrst case we see from x . that the foci the conic are the double points of the system in involution deter mined by the centres Of similitude Of the circles . I n the second case the foci are the points at w hich the centres of similitude Of each pair of circles subtend n a right a gle .

143 . G n tw o w ive circles, sho that there is a single para o m the f bola having d uble contact with the , and that ocus of the curve is the middle point of the centres of similitude of the circles .

144 . G iven four tangents to a conic, to ﬁnd the locus of the centre of a circle of given radius having double contact with the curve .

L a 3 s om o on et , B, y, be the perpendicular fr a p int the four tangents, and let ( I )

’ N if a a be an identical relation . ow , are the perpendiculars f m the ro the pole of the chord of contact and centre of circle ,

c respe tively, on a tangent of the curve we have

2 m ( a where r is the radius of the circle and m is a constant hence from (1) we obtain f “ﬁ t Q x/ ( a x/ (ﬁ ( ) 92 CIRCLES HAVING DOUBLE CONTACT WITH THE CURVE .

for f the equation O the locus . This equation when cleared of

of x m radicals is found to be the si th degree , as ter s of higher

I t w orders vanish identically. can easily be sho n that the locus passes through the circular points at inﬁnity, and the points at inﬁnityon the diagonals of the quadrilateral formed “ 7 w e by the tangents . If w e put 8 in see that the locus Of the foot of the normal at the point of contact of one of the tangents is

32 m 82 z 32 0 W W > W W > m > , w hich being divided by 3 represents a cubic passing through

w 8 is met b a . the points here y , B , y 145 B . y the same method as that employed in the pre

x m h w w w e ceding e a ple we can s o that , hen are given three

b of tangents to a para ola , the locus of the centre a circle of i w v e g ven radius , having double contact ith the cur , is a

e curve of the ﬁfth order . W can prove that this locus is unicursal as follow s - Since there is only one circle of

w given radius having double contact ith a parabola, the co-ordinates of its centre must be capable of being expressed rationally in terms of the coefficients in the eq uation of the b f . B ut i coe fi curve g ven three tangents to a para ola , the cients are quadratic functions of a parameter ; there f & c . ore , 146 . G i iven four points on a con c , to find the locus of the centre of a circle of given radius having double

a cont ct w ith the curve .

L a 8 f m et , B , y, be the perpendiculars ro the points on a

if m the of f line , then Z, , are areas the our triangles f m b i w e the e i e a or ed y the po nts , have id nt cal r l tion

/a 0 M B 217 .

- E V I . CI RCLES CUTTI NG TH CURVE ORTHOGONALLY

AT T O POI NT W S .

148 . There are tw o systems of circles which cut a conic n o orthogo ally at two points, the lines j ining the points being

to of x parallel one the a es of the curve .

2 2 x y I 2 4- s of f 2 i the equation the curve , the equation (72 6

2 C 2 “— — 2 — 2 ’ 2 x + y 2 7 x + a + c e x 0 (1) represents the circle cutting the curve orthogonally at the ’ met b so cc 0 points where it is y the line , and

2 2 6 0 2 2 2 2 ’2 x + — 2 - + b — c + — = 0 2 y y 2 g/ ( ) ?7 represents the circle cutting the curve orthogonally at points 0 on the line y g/ . If r is the radius and a the abscissa Of the centre of the circle we easily ﬁnd

2 8 2 ( r (3)

’ and if r is the radius and B the ordinate of the centre of

2 2 2 2 (6 b ﬁ 0 ) é£ (4)

3 149 . Since the equation ( ) in the preceding example is

w e a c o s t s unaltered if interchange and , it f llow tha the circle I LES TTING HE E ET C RC CU T C URV , C . 95 of the system (1) are also doubly orthogonal to the conic

2 2 a) y 1 = 0 . 2 2 0 6

The points on this conic will always be imaginary when

ur are those on the given c ve real, and real when the latter points are imaginary. It may be observed that the circles of the system (2) are doubly orthogonal to the imaginary conic 2 2 (I?

48 5 o m of Ex . 1 1 0 . The envelope f the syste circles at , is evidently 2 2 2 2 3 2 2 2 x C C 27a c x 0 ( 9 ) , w n m cc - b m hich , bei g transfor ed to elliptic ordinates y eans of the formulae 2 g z w + + c + u 0x w y u , I ,

2 2 3 2 2 becomes (ff v a ) 270 11 0 but this is equivalent to ? 3 8 It v ( 6 O . f m The envelope can also be written in this or , if the v ertices of the curve are taken as the foci of the system

of conics .

151 . I of on x f pairs circles having their centres an a is ,

and cutting each other orthogonally, be described through a x ﬁ ed point on a conic, the circles passing through the vari able points where they meet the curve again have a common

radical axis . The circle having its centre on the axis of x and passing

o x $ on through the p ints 1, 2 the conic 96 CIRCLES CUTTING THE CURVE may be written

2 2 x + y (1)

Now a O x , if is the abscissa f the ﬁ ed point , the circles

2 2 2 2 2 w + e a + C w + e a x 6 0 y ( 1) 1 ,

2 2 2 w + m — b 0 y 8 a 2 , out each other orthogonally if

4 2 2 6 a x a 33 28 a C $ 46 0 2 ( 1)( 2) ( 1 2) , ( ) subject to w hich condition the circle (1) passes through the ﬁxed points determined by the equations

2 2 2 1 + 1) + 6 a

These points satisfy the equation

a 2 2 2 2 2 x + y — 2 — x + a + b a w hich represents the circle cutting the curve orthogonally f at the points lying on x a . The locus o the points (3) is evidently a concentric conic but if the curve is an equilate

x ral hyperbola they lie on the a is of y.

15 I of m Ex . 148 2 . f three circles the syste at , be

w w of dra n through a point , sho that the centroid the three x f corresponding points on the curve lies on the a is o y.

5 of of t m E x . 148 1 3 . P airs circles the sys e at , are described so as to have the same radius ; Show that their centres belong to a system in involution . 2 154 x 0 ortho o . The circle cutting the parabola y p g ’ nally at the points where it is met by the line at x z 0 may be written 2 ' a? x 0 g/ p .

Y CUTTING THE TH G A T . 98 CIRCLES CURVE OR O ON LL , E C

f x 148 of o the system E . show that their points contact lie on the limacon

2 2 2 2 2 2 x cx 6 m 0 ( 9 ) ( y ) . 5 1 9 . L f 0 0 ines are drawn through the ocus , and the centres of circles of the system E x . 148 Sho w that they meet the circles in points lying on the cubic

2 2 + x — — = y (c ) b ( c w) 0 .

160 f . To describe through a point on the curve circles o E x 148 the system . . Elimin ating 9 betw een the equations

2 as y

2 2 C 6

a2 2 2 9“ ’2 — 2 — e x 0 .b + y x

’ d b w w e O and divi ing y x , btain

2 2 ’ 2 6 (x x zb) 2C 0 ;

tw o m ma b hence circles of the syste y be descri ed, and the circles passing through the variable points w here they meet

x m tw x the curve again intersect the a is inor in o ﬁ ed points .

of 48 161 I tw o m E x . 1 be . f circles the syste , de

of m E x 89 scribed to cut orthogonally a circle the syste . Show that the circle passing through the corresponding points on the curve meets the axis minor in the ﬁxed points

2 : g/ a + 0 . ( 99

I — R A O M . VI . N LS

162 . A triangle is inscribed in the conic

2 2 x y 2 a 6 and circumscribed about the conic

2 2 92 9

’2 a

m to S O to show that the nor als , at the vertices f the triangle ,

pass through a point , and to ﬁnd the locus of the point . Writing down the conditions that the sides of the tri angle

“ - m a + 2 ( Bu g ( s) ( a

’ ' ' o S m a b w e should t uch , and eli inating and , obtain

sin (1)

m w a which shows that the nor als al ays p ss through a point . Now the hyperbola

” 2 S E 2 (c wy a a: y) 0

’ ’ of m d w f m w S passes through the feet the nor als ra n ro , y to ;

S m s m hence , since circu scribes triangle circu scribed about ’ S w e m tw o , ﬁnd fro the invariant relation connecting the latter conics 100 NORMALS .

he x m 163 . If in t preceding e a ple

2 2 C 6

2 ’ c 0

m s the locus (2) coincides with S . Thus we see that the nor al to S at the extremities of chords touching the conic

2 2 x y 1 2 6 6 4 ( ) a 6 0 m intersect on the curve . We are permitted to assu e the ’ ’ 1 a b values ( ) for , , as they are consistent with

’ ’ a b

a b

’ the invariant relation connecting S and S .

35 w e of of From E x . see that the locus the centre the

m is circu scribing circle , in this case ,

2 2 2 4 4 $ 9 b C O y 7} .

Also the envelope of the circumscribing circle is

2 2 2 2 2 2 4 4 (w + g/ — a — b ) — a b

E of of From x . 7 the locus the centroid the triangle is

2 2 x 312 1 ( C 2 2 4 C 6 9 0

164 . To ﬁnd the area of the triangle considered in the preceding example .

L et abl « cc- s y/ 1, By/ 2, my, be the ordinate of the vertices of the triangle , then we have

102 NORMALS .

165 A . chord of a conic is a tangent to a parabola which touches the ax es of the curve ; Show that the normals at its

x m of e tre ities intersect on a curve the third order . 66 1 . The circle

2 2 — x + jc/ 2a x

passes through three points on the conic

2 2 w y 2 2 C 6 at which the normals intersect in the same point show that 2 2 62 B2 1

° (B2 (i s 4 6 1 7. From the point w here a normal to the conic

2 2 92 9 “2 62

touches the evolute tw o other normals are draw n to the curve ; show that the line joining their feet is normal to the conic

4 4 61 b 2 2 2 2 ( 1 a} b g/ 4 0

6 T of 1 8 . If S and are the invariants the pencil which joins any point on the conic

2 2 x y “2 62 to of m m x the feet the four nor als drawn to the curve fro , y, show that 3 4 2 2 2 2 3 8 (C ( 6 33 b y ) 2 2 4 2 2 “6 0 x y

69 I f m on lac m n 0 m s 1 . fro points the line y nor al ar e draw n to the conic NORMALS . 103 show that the poles of the chords joining their extremities form a quadrilateral inscribed in the cubic

so 170 . A conic circumscribes a triangle that the normals at the vertices pass through a point ; to ﬁnd the locus of the centre of the curve . If 9 is the angle which the chord joining the points a m w m s , B akes ith the dia eter bi ecting it , it can be easily show n that cot 9 Sin a ) ; w e see 22b ( B hence that the

bs 2 . Com Ar t. 31 E x 10 m condition ( , ) that the three nor als should meet in a point can be w ritten in the form

0 0 0 0 1 cot 1 cot 2 cot 3 , ( )

0 0 0 w of the where 1 , 3 are the angles hich the sides triangle d m B ut m k w i m. a a e ith the ia eters bisect ng the if , B , 7 are

d m of the perpen iculars fro a point on the sides the triangle , we hav e — B sin B 7 sin C + a sin (B C) OOt 01 2 a sin B sin C

m 0 0 m and si ilar values for 2, hence, fro we obtain

5 2 — — w rw O a > “ B) <2> sigA m B an d which represents a cubic passing through the vertices Of the

d of i triangle , the centroi , and the centres the c rcles touching the sides . It may be observ ed that if conics be inscribed in the

so x m s triangle , that the a is ajor passe through the centroid , the foci w ill lie on the cubic 104 NORMALS .

1 1 7 . In the preceding example show that the locus Of the points through w hich the normals pass is the cubic

2 2 2 2 (cos A cos B cos C) a (B 7 ) + (cos B cos A cos C) B(7 a ) 2 2 0 (cos C cos A cos B ) y ( G B ) .

n 172 . To ﬁnd the condition that the normals at six points O 2 2 $ — 1 0 b m . the curve 2 , should be all touched ythe sa e conic a, z Expressing that the normal w hose equation is

by 2 0 . C cos 4) Si n (1)

b w e an touches a conic given y the general equation , obtain equation of the eighth degree in tan betw een the roots Of w hich we ﬁnd three relations by eliminating the constants in E m i f m the equation of the conic . li inat ng ro these relations

of two the roots, we have the condition required R S 0 PQ , w — — here or os os) ,

m) ,

R 2 cos s ( S 2s 2 sin ) ( ( pl ) , sin (q1 pg) ,

28 being equal to S o. 2 If the normals at six points on the parabola y p x 0

b m i w e are all touched y the sa e con c, ﬁnd

By 0 .

m w f m P 173 . Three nor als are dra n ro a point of the conic

2 2 x

2 2 ( 4 f) to the curve ; Show that the product of their lengths is equal to 2 2 2 C 6

w here p is the radius of curvature at P .

106 N ORMAL S .

The curve referred to the asymptotes being written in the form 2 Qxy a O; ( 1)

the hyperbola which passes through the feet of the normals ’ ’ m x fro , y to the curve is

2 x - g/

S x m 1 m m n olving, then , for fro ( ) and and re e beri g ’ ’ 2 Zx 61 that y , we ﬁnd

m 3 and, si ilarly, y

Thus we see that but one real normal can be drawn . MW

VI I I —LI NES MAKI NG A CONSTANT ANGLE W I TH . T E RVE H CU .

1 of m 78 . To ﬁnd the locus the poles of lines aking a constant angle with the conic

2 2 x y 1 ' 5 r

I v m f is the eccentric angle at a point on the cur e , and

n the ma the tangent of the given a gle , equation of the line y be written

a; (b cos q) ma sin <1) mb cos

2 ab mo sin 0 1 ( gb cos (p) . ( ) Comparing this equation with

’ ’ xx g/y 62

l s b c s + ma sin i o ¢ ¢ 2 z ’ ( 0 ab mo sin cos (p

’ y a sin ¢ — mb cos

m ) m hence, eli inating q, and o itting the accents, equation of the locus is

4 2 " 0 513 y x fi 2 mc x 7 1 1l 6 6 6 6 ? ( 1 6 ( L 6 ( if b \ i z o , 108 LINES MAKING A CONSTANT w hich represents a quartic having a node at the origin and

of two nodes at inﬁnity. We see thus that the envelope the 1 line ( ) is a curve of the sixth order w ith six cusps . 9 17 . To ﬁnd the condition that three lines making an angle tan“1 m with x2 2

h 2 2 a 6

a m . at the points , B , 7 should eet in a point - 1 Since the equation of the line making the angle tan m ’ ' w w ith the curve at the point , y is

’

ar ’ z ’ z ' ' m a m b x c w 0 ( z y y y) , b?

o b 8 a C m s Ar t. Ex 1 1 1 . it follows that (see , , ) the hyperbol

’ z z ' z ' z ’ e 2 S as m( c xy b yx a x y) b w x (r g/y a O (1)

on S w m passes through the points , at hich lines aking the “1 ’ ’ tan m w w angle ith the curve intersect in , y ; hence , if P, Q, o h m are a pair f lines passing t rough the latter points , we ust have ’ A 5 S S APQ , m where, in ter s of the eccentric angles

I a + cx — ( B) ( s) , Q

— — Q coa l 8) + %sin 8) cos (y

2 E cc -efﬁ cients 03 m so quating, then , the of , y, y, and the ab lute m ter s in this identity, we obtain

cos é

2 mo ab = >x sin ( a + 6 y

i 2 - a l) k A cos % (a B) cos ﬂ y

110 LINES MAKING A CONSTANT m and circu scribed about the concentric conic, whose g ential equation is

2 2 E AY + 5 1 ZE 0 7 N; ,

39 m 3 E x . we have fro ( ) and , / 2 H 4 5 cos s1n C0 z 2 ¢ 1 , Ka 6 ab

2B cos gb Cg ab 2B c; cos ( hence p sin ) 9 p ’ abC

— = but p siu ¢ qcos ¢ sin (IS + sin (7 + a ) + sin ( a + B)

m E x 179 w e . m thus see at once fro , that lines aking the

"1 constant angle tan w ith S at the vertices of the tri angle pass through a point . To ﬁnd the locus of this point w e express the invariant E 1 9 x . 7 m condition that the hyperbola , should circu scribe triangles circumscribed about 2 w e thus ﬁnd

2 2 2 2 2 4 a a mx as m 2174 6 mm a: m m 0 = 0 (y ) ( y) (y ) ( y) ,

? ’ 2 ’ w here air b y 212 xy 1 O

i of 2 x cc - is the equat on in , y ordinates . 184 L m . ines aking a constant angle with a conic at the vertices of an inscribed triangle pass through a point on the curve ; show that the locus of the centroid of the triangle is a concentric conic . 85 1 . From the point of intersection of two lines making

‘ 1 a constant angle tan m w ith the conic 2 2 x y T ANGLE WITH HE CURVE . 111 at the extremities of a chord which passes through the ﬁxed ’ ’ x w w m the point , y , the two other lines are dra n hich ake same angle w ith the curve show that the chord joining the feet of the latter pair of lines touches the parabola whose tangential eq uation is

2 2 ? 2 z 2 z ' 2 6 $ ) a d 5 a A mo v 6 A O . 61 ( ( 71 1 y ) . z “)

86 of m 1 . The locus of the intersection lines aking a con stant angle w ith a conic at the extremities of a chord which passes through a ﬁxed point is a curve of the third order (see 3 0 Combs Ar t. 7 I x on m , , f the ﬁ ed point is the dia eter w hich cuts the curve at the given angle , the locus reduces to m a conic , as the dia eter in this case is part of the locus . The

x locus also reduces to a conic if the ﬁ ed point is at inﬁnity. m m 187 . Fro the point where a line aking a constant angle with the conic touches its envelope the two other lines are draw n which make the same angle with the curve show that the line joining their feet cuts a concentric conic at

a constant angle . 88 A r m x so 1 . conic ci cu scribes a ﬁ ed triangle , that lines making a given angle 0 with the curve at the vertices pass through a point ; show that the locus of its centre referred to the triangle is the cubic

2 2 2 z 7 z 2 (6 7 > 0 a l (a 6 ) 2 0 0 1396167 0 sin C

s e Ex ( e .

f 517 189 . I a 8 , B , 7 , are constants, and , y rectangular

cc- ordinates , show that the line

cuts at a constant angle a conic having the origin for centre .

Also show that this cannot be the case if a B . AK G A TA T A G T 112 LINES M IN CONS N N LE, E C .

‘ l 190 . If three lines making the given angle tan m with z 4am 0 th e 1 the parabola y at points 3 1 , yz , y, pass o thr ugh a point , we ﬁnd

+ 9 1 yz + ys

Now

is the condition that four points on the cur v e should lie on a circle ; hence we see from that the circle passing thr ough the three points meets the curve again in the ﬁxed point

2a m

9 S ir m 1 1 . Triangles are inscribed in a parabola , and c cu ’ scribed about a parabola S ; show that lines making a cer tain constant angle w ith S at the vertices of one of the

w triangles pass through a point , and sho that the locus of hi t s point is a right line . 9 on the 1 2 . If two lines are drawn through the point g/ z 4b 0 m parabola y , to eet the curve again at the angle

‘ l tan m of , we ﬁnd that the equation the line joining their feet is ’ ’ 4mm): (my 2a) y 2a (y 4mm) 0

’ x which , when y varies, passes through the ﬁ ed point

1 2m2 ( ) .

The locus of this point for different values of m is the equal parabola = 0 .

A I 1 14 OSCUL TING C RCLES . We thus obtain the equation

z 2 - sin egB 1 + v 1 5 0

B ut this is evidently satisﬁed 1 1

’ it ’ cos B cos C cos C cos A cos A cos B and these are evidentlythe cc-ordinates of the fourth common

e tangent of 2 and the circl inscribed in the triangle . The s mm three osculating circle have , therefore , this line for a co on tangent . ’ and 2 are s os s If 2 not both ellipse , the culating circle are touched by the fourth common tangent of 2 and one of the exscribed circles . The above result can be readily proved by means of ’ f t b C m x m elliptic unc ions ; for , y hasles s theore , the e tre ities of the diagonals of the quadrilateral formed by the common tangents of a conic and a circle lie on a confocal conic ;

m E x . 44 s ou hence , fro , we ee that if four tangents are t ched by the same circle we must have

4m or K .

Now three tangents coincide at the point of contact of an u u u o osculating circle hence , for the points of contact l , z , s f s u three o culating circles which touch the tangent , we have

1 - - w u 10 u 11, If u 1 71 K hence l £ , and , 3 , s 3 g ,

m w u u u u fro hich it follows that the tangents l , z , , are to ched

by a circle touching u. There are nine osculating circles OSCULATING CIRCLES . 115 t six s m ouching a given tangent, but of the e are i aginary, cor responding to the imaginary periods of u. 9 1 4 . S w u ho that but one real osc lating circle , besides the

to one at the point of contact, can be described touch a given tangent to a parabola .

9 ix v 1 5 . S osculating circles of the cur e

2 2 at s 2 2 a b are described to cut orthogonally the circle

to show that their centres lie on a conic.

Ex ul w x pressing that the osc ating circle hose centre is , y

r S i and radius cuts orthogonally, we obta n

— 2 = r 0 . (1)

B ut m Combs A rt. fro the equation of the osculating circle ( , 2 3 51 E x . v ms , ) we ha e, in ter of the eccentric angle,

6 b

2 2 2 2 2 2 2 ? 2 x y r (o 26 ) 0 0 8 9 (b 2a ) sin 0 (2)

z z z z 4 2 2 therefore a z b y 0 ( 1 3 sin 0 0 0 8 0) hence from ( 1) and (2) we obtain

2 2 z g 2 z z My o) (a b ) (29x 2f y c) 3 (a o: b y )

4 4 z z + a + b - a o = 0 3 , ( ) which proves the theorem. I f 3 the curve is an equilateral hyperbola, the conic ( ) has double contact with the curve . 196 2 . If four osculating circles of the parabola y px 0 1 2 1 16 OSCULATING CIRCLES .

b m are cut orthogonally y the sa e circle , show that the ordi nates of the points of contact are connected by the relation 1 2 __ y

z 197 a: 0 . Three circles osculate the parabola y p at the points sho w that the equation of the circle cutting them orthogonally is 1 2 2 2 9 9 — 19 (w 31 ) 7 ( 1 2 ma p s) w 511 19 1 1 2 p s) 3/

l 2 — + 7 ( p g p 1 p3)

9 B 9 E f w here 1 1 y 1 2 ra/ z , p a ro g/z yg

I f 198 . the osculating circle of a conic cut the director

9 w cos 0 w circle at an angle , sho that here p is the per pendicular from the centre on the tangent at the point of k f o . contact , and is the radius the director circle

he 199 . Let t tangent to a hyperbola at a point P meet m A B if m the asy ptotes in , perpendiculars to the asy ptotes A B m m P A ’ B ’ mi at , eet the nor al at in , , show that the ddle A ’ B ’ point of , is the centre of curvature at P . 0 2 0 . The tangent to a conic S at the point P meets a m m S ’ A B concentric , si ilar and si ilarly situated conic in , ; if o m S ’ A B m m S P the n r als to at , eet the nor al to at in ’ ’ ’ ’ A B m A B of , , show that the iddle point of , is the centre

curvature at P .

s 201 . If S is the quare of the tangent draw n from the point whose eccentric angle is 0 on the conic

2 £6

1 18 OSCULATING CIRCLES .

c ally. The result is then div isible by the square of the

m mm S S S r e distance fro the point co on to I , 2, 3 , and the

maining factor gives the equation of the curve . 2 02 . Three osculating ci rcles of a conic pass through the

s m ur if ( i a e point on the c ve ; p, x, t are the angles at which they r A B C m b inte sect, and , , the angles of the triangle for ed y

of c S w their points ontact , ho that

— — = 3A 7r = 3B 7r = 3 0 ¢ , ¢ , x

20 t 3 . Three oscula ing circles of a conic pass through the same point on the curve ; show that the triangle formed by their centres is similar to the triangle formed by the extr emi ties of the diameters conjugate to those drawn to the points of contact . Also Show that the ratio of corresponding sides is equal 2 a 62 b - w a m x of the u . if here , are the se i a es c rve ab X . CONI CS HAVI NG DOUBLE CONTACT W I TH A FI ED NI X CO C .

204 . A conic has double contact with a fixed conic and a x fi ed circle to find the locus of the foci . If the tangential equation of the fixed conic is

2 2 2 2 E E C A + 6 and that of the circle

’ ’ ’ ? 2 2 s (mA y u r (A M)

2 ’ s x 44 2 o e E . we have en in that, when 2 breaks up int £3 2 it of a factors , is a root the cubic equ tion

’Z y + 2 2 2 a — la 5 -4 ; s

? it h 1f i 2 2 E F ence we wr te , 172 where E and F are the extremities of one of the diagonals of the quadrilateral formed by the common tangents of E and the equation h 2 2 ’ 2 0 E + 20 2 2 + F = 0

see Combs Art. 287 o con ( , ) represents a c nic having double tact w ith 2 and Writing this equation (3) in the form

? 2 ( as F 49 a /\ t 4ar A 0 ) ; ( a ( r) , we see that the points

- 1) = o 120 CONICS HAVING DOUBLE CONTACT

. of w e are foci Taking , then , the envelope this equation

of f i have , for the locus the oc ,

E F c s,r ,

? 2 2 m 2 2 h A 1 0 or fro ( ) ( 1 ) , which represents the confocal conic passing through the

E e E x 44 2 points and F ( se . ( )

of w m This conic gives the locus the foci , hen the ajor axis of the variable conic passes through the centre of for the points represented by the equation (4) are evidently / ’ (If A 1 0 w i n collinear with yu , h ch represents the ce tre of W hen the minor axis of the variable conic passes ’ ’ E f i i - S m through the centre , the oc are the ant points ( al on s

Hi her lome Cur ses Ar t. 139 of tw o 5 g P , ) points on the conic ( ) ’ ’ w i w x co . ﬁn hich are coll near ith the ﬁ ed point , y To d the locus in this case , let

1 E S 0

be of 5 as cc - m th the equation ( ) in , y ordinates ; for ing , en , the equation of the chords of intersection of S and

(93 9 02 (y W 0

mbs Ar 3 0 = Co t. 7 x (see , , and e pressing that this equa

b cc - of x w e tion is satisﬁed y the ordinates the ﬁ ed point , obtain the locus required

7 1 r 2 2 x 2 I I r 2 { sw ay— y) 1 + W 2“ 1 (x + y — a — boucc— m o- o) s

" 1 Z O

122 CONICS HAVING DOUBLE CONTACT

centre Of J are evidently determined by the equation

hence we may w rite (1)

2 2 2 ’ 772 t 713 (A It ll 1) (fvA W ) W 0 (2)

x o o of N where , y are the co rdinates of ne the foci . ow, if this conic (2) touch the conic

2 2 2 2 2 7733 v k A m u v 0 ( ( H) } { ( l yu ) } ,

m Co bs 8 m Art. 3 7 we ust have ( , )

2 2 2 2 ” m a? x 772 { ( ( 9 ) 70 (1 M 0} { 0721 ( 1 W ) k ( 1 10 }

m 1 — k2 2 ( )(ml l )(xx1 jog/ 1) (1 j: mm} 0 ; but this relation can be written in the form

m 1 2 2 2 + 2 2 x y + k + x1 + y1 + k m 1

2 z i $ r 11 (y yxl l m

2 tw o x s m hence , if the conic ( ) touch ﬁ ed conics of the yste , we m have , fro 2 2 0 S i 3 (P1 i 1 21 (x/ r x/ 2 1 , m 2 l S $ S where I 1 yf 2 & c .

B ut s 4 of f O thi equation ( ) represents a pair con ocal conics , f w h of one of x ich each focus is also that the ﬁ ed conics .

of of x Taking account , therefore , all the foci the ﬁ ed conics , we see that the locus consists of four pairs of conics whose foci are formed out of the foci of the ﬁxed conics . These four pairs Of conics give the locus Of the foci when the major

x n Of a is of the variable co ic passes through the centre J . ) Vhen the minor axis of the variable conic passes through WITH A FIXED CONIO. 123 the of J - of I m i centre , the foci are the anti points the ag nary

ar e J . I w x foci which collinear w ith the centre of f i g/ 1 , z yz

o f Of x are the c ordinates o the latter pair foci , and , y those of a real focus, we have (5)

B ut x s if 1 ml is a focus of the conic it is easily hown that the other focus is 1 n 1 m

"2 2 "3 and if (a: o ) (y B) (mo 4» by o)

e of x m 3 is the quation the locus of l , y1, we have , fro ( ) m 2 2 ( x + + k 1 y1 m i ? m’ 1 2 2 ( ) + 15 where ’ B m 1 hence i s equal to an expression of the form

’ ’ ’ a x b / c l g l . m We have then , fro

-1 2123 331

2 1 + x - 1 1 y y1 J a (

' ’ ’ a x b o where y l g/ l , m b and i , y, are connected y the relation

2 x ax 5 + ( 1 (3 h [31 ( , 9 1

b m m m We thus ﬁnd, y eli inating i , y, fro these equations , that

of 93 . t the locus , g/ is a bicircular quartic There are eigh 124 CONICS HAVING DOUBLE CONTACT s s ff uch quartic altogether corresponding to the di erent conics . The entire locus thus consists of eight conics and eight

bicircular quartics . 209 A . n x conic has double co tact with a ﬁ ed conic, and touches two ﬁxed parallel lines ; Show that the envelope of the m s of — S m asy ptote consists two conics concentric , i ilar, and

Similarly situated w ith the ﬁxed conic . 2 0 A ﬁx f 1 . conic has double contact with two ed con ocal

conics ; to ﬁnd the locus of its foci . Let 2 2 2 2 2 5 a A + u ~ v = 0 s ,

’ Q 2 3 2 2 2 E A 6 11 v 0

of x be the tangential equations the ﬁ ed conics , and let

z ’z 2 2 2 — = — 6 = h a a o ,

2 2 2 3 z ’ 2 2 13 0 Then 0 h ( ck v ) 20 01 2 b x ) + ( A v ) 0 (1)

represents a conic having double contact with E and The axis of a: is one of the principal axes of the conics of this

system. ombs A 258 w e C rt. To ﬁnd the foci of have ( , )

— 1 2 Gx + A — B =- 0 ) ,

y ( Caz G) 0

2 2 2 ’g C 11 0 1 2 b b 0 where ( ) ( ) ,

2 2 z ? 2 ’z G 011 ( 0 A B = o lz (0 1) -1 o ) 0 ;

m 0 m 0 hence , o itting the factor y , and eli inating between

2 Ca: G 0 the equations ( ) and , we get

0 — z 0 C ,

126 CONI CS HAVING DOUBLE CONTACT

- 212 . The differential equation in elliptic cc ordinates

( in dv 0 1 2 3 ’ 2 2 ( ) { (a M00 a x/ Ma 1, )( v represents a system of conics having double contact with the ’ o a o b of confocal c nics )u , for, y the theory elliptic O t s 1 in functions, the integral f hi equation ( ) can be written either of the forms

2 z — 2 - u ) (w v ) l

' ’ A 65 0 0 Mv 0W ) l , m A B &c . n where , , , are co stants ; and if we transfor these

x 2 C co- e pressions ( ) to artesian ordinates, we have

; 2 N a row v » ab 1

” (r — a (1

f m 1 hence , ro we see at once that ( ) represents a conic

x c a having double contact with the ﬁ ed onics . This equ tion (1) belongs to the system of conics whose equation is given 2 0 m m 1 . at E x . In a si ilar anner we can Show that the differential equation

o s o m s o E x 210 belongs t the ec nd syste of conic c nsidered in . . A S gain, we can how that

d“ 031) s — - — u — w w c maz) (a mi M o r ns vac 4 » WI TH A FIXED CONIC . 127 is the differential equation of the system of conics at

. 2 0 Ex 1 . From these differential equations we see that the two

of m can conics each syste , which be drawn through a point, make equal angles with the conics confocal w ith the given ones that pass through the point . A if b n o o o lso , g is the a gle between the tw c nics f the system we have

2 2 0 0W ) ( v - Ira11 5 C)

a S m o m nd i ilar values for the ther two syste s .

213 . If the normal at a point P of a conic Of the system

4 x m a S o ( ) in the preceding e a ple touches the conic p , h w the locus of P is the bicircular quartic whose equation to rectangular axes is

2 2 2 ’2 ? 2 ’2 2 2 (co y) (b 2a ) x (a 26 ) y

“2 2 ” ‘ a a a o 0 .

a he s s Ex 212 214 I o o t m . tho . or o f c nic f y te , cut g

a a of s m 4 s m x m s n lly conic the yste ( ) in the a e e a ple, how that the locus of their intersection consists of the tw o circular cubics " 2 2 2 f 2 9 l r b 090 1 o ( 2 ( a/ 0 . r. : 0 y ) i .

5 s m Ex . 2 2 21 . I o s o 1 u f two c nic f the sy te , o t each

s o s s other at right angle , show that the l cu of their inter ection is the bicircul ar quartic

2 2 2 2 ’z 2 f ’2 z ’2 av a a a: b b a a b b ( ( ) ( ) y 0 .

6 m the f x 21 . Fro oci of the ﬁ ed conics tangents are drawn 4 Ex 2 2 to the systems of conics ( ) . 1 ; Show that the 128 CONICS HAVING DOUBLE CONTACT

o o o res ec equati ns of the loci of their p ints of c ntact are , p tiv el y,

2 2 x (0 1

2 2 £19 y

2 2 2 ’2 o y b b 0 .

N m om o of x 217. or als are drawn fr the f ci the ﬁ ed conics to the systems of conics referred to in the preceding example ; o Of show that the l ci their feet are , respectively,

2 2 2 x + y — a

2 2 2 2 x (x + g/ — a

2 2 2 2 ’2 2 2 ’2 2 2 ’2 a x C b a 0 (x y ) (C ) ( ) g/ a .

N m w om of x 218 . or als are dra n fr the centre the ﬁ ed conics to the systems of conics Ex . 212 ; Show that

of f the loci their eet are , respectively

2 2 + g/2 ) (a

’2 2 + b ) y

9 S of of m 21 . how that the locus the vertices the syste of 2 o E x . 2 1 c nics , is

2 902 + y

2 ’2 2 f 2 + a a x + [ l/

of s 220 . Show that the loci the vertices of the system o x 2 2 E . 1 c nics , are , respectively,

2 ’2 z f z 2 (0 b ) _ b b 0 ) O

2 4 2 2 2 ’2 2 2 0 95 0 513 (a a am (y 0 ) z . . 0

CONICS HAVING DOUBLE CONTACT

22 n x m the a 4 . I S n the precedi g e a ple , how that vari ble director circle has double contact with one or other of three

of x bicircular quartics , which reduce to cubics if one the ﬁ ed conics becomes a parabola . 225 x . A parabola has double contact with a ﬁ ed conic ,

x of x and touches a fi ed line ; to find the envelope its directri . Let the tangential equation of the fixed conic referred to its axes be 2 2 2 2 o k 6

2 2 2 2 2 2 a )\ + 0 11 v ( aA B)“ v ) 0 represents a parabola having double contact w ith

N x 2 ombs Ar t 294 C . ow, the directri of ( ) is ( , )

— 2 2 a x + 2By ( a +

2 the x la: m n 0 and if ( ) touch ﬁ ed line y , we have

2 2 2 2 la m n C 2 6 m 0 4 B i ) . ( )

B ut the envelope of subject to the condition is found to be

2 2 2 2 2 2 2 2 2 2 2 ( Z m )(oo + y a 6 ) —{ lx + my n i a l — b m ) which represents two parabolas hav ing double contact w ith the director circle Of the ﬁxed conic

226 . I x m S f n the preceding e a ple , how that the locus o the foci of the parabola consists of two circular cubics with Ex 8 nodes (see . 7)

227 . An equilateral hyperbola has double contact with the ﬁxed conic WITH A FIXED CONIC . 131

' ’ and ss s h x x s pa e t rough the ﬁ ed point , y ; show that the locu of its centre is the bicircular quartic

2 2 2 ? 2 z z ’ ’ 2 1 4 6 (x y ) ( a b ) (b xm an ) }

z g 2 z z 4 2 4 z (d e g/ a b )(b x a g ) O .

228 . Show that an inﬁnite number of equilateral hyper bolae can be described to have double contact w ith the ﬁxed conics whose equations to rectangular axes are

2 2 x y “2 b2

? z z 2 ? 2 2 ? 2 2 c z ar b y 2 (a b ) ( ax + ﬂy) (a 6 ) ( a B ) (6 6 O .

229 An c w . equilateral hyperbola has double ontact ith 2 b 4mm O x the para ola y , and passes through the ﬁ ed point ’ ’ w o , y ; show that the locus f its centre is the conic

2 2 2 ’ ? 2 47 0 (g/ 7195) 4m ) (yg/ 27m: 2mm 4m ) . ( 132 )

II I — RELATI ONS OF A CI RCLE AND A C C ONI .

280 A 8 s . circle intersect a conic in four points ; to show that perpendiculars at these points to the radii vectores draw n ’ from a focus are all touched by a circle S . R n one u eferri g the conic to of its foci , its polar eq ation is

r (1 6 0 0 8 9) = l ; and if the equation of S is

? 2 r 2r a cos B + sin 9 79 O 2 ( 6 ) , ( )

h m r 1 we ave for their intersection , eli inating between ( ) and

2 2 1 — 2l 1 e cos O a t) sin 0 1 e cos tl O ( ) ( cos B ) ( ) ,

2 2 2 2 g or P k (16 8 2Zea ) cos 0 2 (lo e Ia ) cos t}

2Z in B 216 t) t) B s 6 sin cos O . (3)

N o of ow, the equati n the perpendicular to a focal radius vector at its extremity is

33 0 0 5 0 ysin ﬂ r and if this line touch the circle

’ 2 2 ” a: a 1 0 ( ) (y 33 ,

AT F A AND A 134 REL IONS O CIRCLE CONIC .

The equation of the conic being

the equation of a tangent is

z z w oos w + y sin w b sin w ) = 0 ; (1) then if p is the perpendicular on this tangent from the ﬁxed d pomt x g/ w e know that i s the reci procal of the common

’ ’ of x tangent the conic and a circle whose centre is , y , and

s . mm n radiu p Hence , considering the four co on ta gents of

) m the conic and circle , 1 is the sa e for all , and we have 1 d (2) 1, (1p

B ut if w e express that the line ( I ) touches a circle w hose ’ ’ x w e centre is , y , and radius p , get

’ 2 z z x OOS w g/ siu m p ¢ (a (30 8 0 ) b sin w ) O ; (3)

m n ew z m and putti g , then , in this equation it beco es

’ ’ 2 ' ’ “ m 3 2 2 12 x + — I z O { ( y/ ) 1 yM f s ,

u 4 2 aq 0 : o 4 M mm ( , ( )

m w e 2 u) 4 ) fro which readily deduce that g , where

'z 3: y c

Em is of see f m 2 hence , since independent p , we ro ( ) that I t

I f the c u i b n tw o cir le to ch the con c , y drawi g tangents to LAT N F A AND A RE IO S O CIRCLE CONIC . 185 the u v n ﬁ w e c r e i de nitelynear one another , obtain the follow in mm t 16 g values for the lengths of the co on tangents l , 2,

‘ t r ( 19 0 0 ” . (p ) tr p W ,

1 I t [ 1 dd s 55 0) ( 9 cot a

h r of v w ere is the radius of the circle , p the radius cur ature of of ( the conic at the point contact , and p the angle which the diameter of the conic at the same point makes with the 1 b to im — i curve ; hence , y proceeding the l it , in this case 251 if; 2p COMP is b replaced y ” (9 T)

233 T o m n . show that a circle eets a co ic at angles the

- z sum of w hose cc tangents is equal to ero . I f a circle of ﬁxed centr e and variable radius r meet the

an 1 v curve at angle p, we ha e cot ¢ w here w is the angle which the radius makes w ith a ﬁxed

i for o f r l ne ; hence, the four points intersection , since is the s m for a e all ,

2. cot t g 22;

B ut since a pair of chords of intersection of the conic and

e ax s circle are qually inclined to an i of the conic, it easily

w 2 m m follo s that is constant ; hence , fro

c t 0 E o ¢ .

I f ir the i the sum tw o cc- n the c cle touch con c , of ta gents 2 21' cot

x e meaning as in the pr eceding e ampl . R AT F A A 136 EL IONS O CIRCLE ND A CONIC .

2 4 3 . If E is the sum of the squares of the lengths of the six chords of intersection of the conic