EllipsesEllipses andand HyperbolasHyperbolas In this chapter we’ll see three more examples of conics. In this chapter we’ll revisit some examples of conies. EllipsesEllipses 1 If youIf beginyou begin withwith the unitthe unit circle,circle.C ,C’. andand youyou scalescalex-coordinatesr-coordmates byby somesome nonzerononzero numbernumbera, anda. youand you scalescaley-coordinatesj-coordinates byby somesome nonzerononzero numbernumber bb,, -

the resultingthe resulting shapeshape in thein planethe plane is called- is called an anellipseellipse.. on on

IoI I-o I I-o ‘I I II

Let’s begin again with the unit circle ‘I II C’, the circle of radius 1 centered at Let’s begin again with the unit circle C1, the circle of radius 1 centered at (0, 0). We learned earlier that C’ is the set of solutions of the equation (0, 0). We learned earlier that C1 is the set of solutions of the equation x-+y9 2 =1 x2 + y2 = 1 The matrixa 0D scales the x-coordinates in the plane by a, and The matrix scales= ( the) x-coordinates in the plane by a, and it scales it scales the0 y-coordinatesb by b. What’s drawn below is D(C’), which is the shape resulting from scaling C’ horizontally by a and verticallyaby 0b.It’s an the y-coordinates by b. What’s drawn below on the right is (C1), example of an ellipse. 0 b which is the shape resulting from scaling C1 horizontallyH’ by a and vertically by b. It’s an example of an ellipse. op

0 op

Using POTS, the equation for this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix op _ z:3 -co op _ z:3 D1=(j -co II + +

?) II + + 1874 4 0 0 II II +

204 + - - on on - I-o I I-o ‘I I II ‘I II on I-o I

‘I a 0 Using POTS,II the equation for this distorted circle, the ellipse (C1), Ellipses and Hyperbolas 0 b 1 2 2 is obtainedIn this bychapter precomposingwe’ll revisit the equationsome examples for Cof,conies. the equation x + y = 1, by the matrix Ellipses −1 1 a 0 a 0 If you begin with the unit circle. =C’. and1 you scale r-coordmates by some 0 b 0 b nonzero number a. and you scale j-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse. op op op _ z:3 -co op _ z:3 -co op II + IoI+ II + + Let’s begin again with the unit circle4 C’, the circle of radius 1 centered at 4 0 0 (0, 0). We learned earlier that C’ is the setII of solutions of the equation II

9 2 +

x-+y =1 + op _ z:3 The matrix D scales the x-coordinates-co in the plane by a, and II + + =

( ) 4 it scales the y-coordinates by b. What’s drawn below is D(C’), which is the 0

shape resulting from scaling C’ horizontallyII by a and vertically by b. It’s an

example of an ellipse. + H’ 1 a 0 x y Because 1 replaces x with a and y with b , the equation for the ellipse 0 b a 0 (C1) is 0 b 0 x2 y2 + = 1 Using POTS, the a b which is equivalent to theequation equationfor this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix x2 y2 2 + 2 = 1 a D1=(jb ?)

187 The equation for an ellipse described above is important. It’s repeated at the top of the next page. 205 Ellipses and Hyperbolas Suppose a, b ∈ − {0}. The equation for the ellipse In this chapter we’ll revisit some examples ofR conies. obtained by scaling the unit circle Ellipses by a in the x-coordinate and by b in the y-coordinate is If you begin with the unit circle. C’. and you scalex2r-coordmatesy2 by some + = 1 nonzero number a. and you scale j-coordinates by asome2 bnonzero2 number b, the resulting shape in the plane is called an ellipse.

IoI Let’s begin again with the unit circle C’, the circle of radius 1 centered at (0, 0). We learned earlier that C’ is the set of solutions of the equation x-+y9 2 =1 Example. The matrix D scales the x-coordinates in the plane by a, and x2 y2 = • The equation for the ellipse shown below is + = 1. In this ( ) 25 4 it scales the y-coordinatesexample,by web. areWhat’s usingdrawn the formulabelow is fromD(C’), thewhich top ofis thethe page with a = 5 and shape resulting fromb = 2.scaling C’ horizontally by a and vertically by b. It’s an example of an ellipse. H’

0 tn

Using POTS, the equation for this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix ************* D1=(j ?)

187

206 Ellipses and Hyperbolas

In this chapter we’ll revisit some examples of conies. Ellipses If you begin with the unit circle. C’, and ou scale i-coordinates by some nonzero number a, and you scale y-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse. C, b

r

Let’s begin again with the unit circle the circle of radius 1 centered at (0. 0). We learned earlier that C’ is the set of solutions of theEllipsesequation and Hyperbolas 9 9 In this chapter we’ll revisit some examples of conies. Circles are ellipses The matrix D scal:s th:x-coordinates in the plane by a, and A circle is a perfectly round ellipse. Ellipses = ( ) If weit scales scalethe they-coordinates unit circle byby theb. What’s same numberdrawnIfbelowryou > begin0is inD(C’), bothwith which coordinates,the unitis thecircle. C’, and ou scale i-coordinates by some thenshape we’ll stretchresulting thefrom unitscaling circleC’ evenlyhorizontally in all directions.nonzeroby a andnumberverticallya, byandb. youIt’s scalean y-coordinates by some nonzero number b, example of an ellipse. the resulting shape in the plane is called an ellipse. C, C’ b Ellipses and Hyperbolas r In this chapter we’ll revisit some examples of conies. Ellipses If you begin with the unit circle. C’. and you scale r-coordmates by some Using POTS. the equation for this distortedLet’scircle,begintheagainellipsewithD(C’).the unitis circle the circle of radius 1 centered at nonzero number a. Theand resultyou scale willj-coordinates be another circle,by some onenonzero whosenumber equationb, is obtained by precomposing the equation for C1,(0. 0).the Weequationlearnedi2earlier+ y2 =that1, byC’ is the set of solutions of the equation the resulting shape in the plane is called an ellipse. the matrix x2 y2 9 9 + = 1 rD’=(E52 r2 We can multiply both sides of this equation) byTher2 tomatrix obtainD the equivalentscal:s th:x-coordinates in the plane by a, and 187 = ( ) equation it scales the y-coordinates by b. What’s drawn below is D(C’), which is the 2 2 2 x + y = r shape resulting from scaling C’ horizontally by a and vertically by b. It’s an which we had seen earlier as the equation for Crexample, the circleof an ofellipse. radius r centered at (0, 0). IoI C’ Let’s begin again with the unit circle C’, the circle of radius 1 centered at (0, 0). We learned earlier that*************C’ is the set of solutions of the equation x-+y9 2 =1

The matrix D Hyperbolasscales the fromx-coordinates scalingin the plane by a, and = We’ve( ) seen one example of a hyperbola, namely the set of solutions of the it scales the y-coordinates by b. What’s drawn below is D(C’), which is the 1 Using POTS. the equation for this distorted circle, the ellipse D(C’). is shape resulting fromequationscalingxyC’=horizontally 1. We’ll callby thisa and hyperbolaverticallyHby ,b. theIt’sunitan hyperbola. obtained by precomposing the equation for C1, the equation i2 + y2 = 1, by example of an ellipse. the matrix H’ D’=(E5 ) 0 187

Using POTS, the equation for this distorted circle, the207ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix D1=(j ?)

187 Ellipses and Hyperbolas

In this chapter we’ll revisit some examples of conies. Ellipses If you begin with the unit circle. C’. and you scale r-coordmates by some nonzero number a. and you scale j-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse.

IoI Let’s begin again with the unit circle C’, the circle of radius 1 centered at 1 (0, learned Suppose c > 0. We can horizontally stretch the unit hyperbola H using 0). We earlier that C’ is the set of solutions of the equation 9 c 0 x-+y 2 =1the diagonal matrix . This is the matrix that scales the x-coordinate 0 1 by c and does not alter the y-coordinate. The resulting shape in the plane, The matrix D scales the x-coordinates in the plane by a, and = c 0 ( ) (H1), is also called a hyperbola. it scales the y-coordinates by b. What’s drawn0 1below is D(C’), which is the shape resulting from scaling C’ horizontally by a and vertically by b. It’s an example of an ellipse. -

H’ - - (c,i) .xI—cx

on D(H)

on 0 on

Using POTS, the equation for this distorted circle, the ellipse D(C’), )

is ,)ço

I-o

I ‘4O

obtained by precomposing = ‘I the equation for C’, the equation x2 1, by I-o I-o II

+ y2 I I 1 o

fj ‘I c\ ‘I II II the matrix =) D1=(j ?)

187 Jo

c 0 The equation for the hyperbola (H1) is obtained by precomposing 0 1 c 0−1 1 0 the equation for H1, the equation xy = 1, with = c . This is 0 1 0 1 the matrix that replaces x with x and does not alter y. Thus, the equation c c 0 1 for the hyperbola (H ) is op op

0 1 op x y = 1 c which is equivalent to op _ z:3

xy = c -co op op _ z:3 _ z:3 -co -co II c + From now on, we’ll+ call this hyperbola H . II + II + + + 4 0 4 To summarize: 4 II 0 0

208 II II + + + Ellipses and Hyperbolas

In this chapter we’ll revisit some examples of conics. Ellipses If you begin with the unit circle, C’, and you scale x-coordinates by some nonzero number a, and you scale y-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse. C) D(c’) Let’s begin again with the unit circle C’, the circle of radius 1 centered at (0, 0). We learned earlier that C’ is the set of solutions of the equation 9 9

The matrix D scai:s the -coordinates in the plane by a, and = ( ) Let c > 0. Then Hc is the hyperbola it scales the g-coordinates by b. What’s drawn below is D(C’), v.hich is the shape resulting from scaling C’thathorizontally is the setby ofa solutionsand vertically of theby equationb. It’s an xy = c. example of an ellipse. I1c

Ellipses and HyperbolasEllipses and Hyperbolas

In this chapter we’ll revisit some examples Inof conics.this chapter we’ll revisit some examples of conies. %____; %____; Ellipses Using POTS. the equation for this distorted circle, the%____; ellipse D(C’). is II Ellipses II 0 0 x x --‘ --‘ II obtained by precomposing the equation for C’, the equation x2 = 0 x 1, by

+ y2 --‘ If you begin with the unit circle, C’, and youIf youscalebeginx-coordinateswith the unitby somecircle, the matrix Example. C’, and you scale i-coordinates by some TT1- ,J,.J, TT1- nonzero number a, and you scale y-coordinatesnonzeroby numbersome nonzeroa, andnumberyou scaleb, ,J,.J, TT1- y-coordinates ,J,.J, by some nonzero number b, 2 the resulting shape in the plane is called an• theTheellipse.resultingD1=( equationshape forin thethe hyperbolaplane is calledH ,an obtainedellipse. by scaling the unit I’ ?) I’ hyperbola by 2 in the x-coordinate is xy = 2. I’ c0 187 c0

D(c’) c0 C) N Let’s begin again with the unit circle C’, the circleC)of radius 1 centered at Let’s begin again with the unit circle the circle of radius 1 centered at (0, 0). We learned that earlier C’ is the set (0,of 0).solutionsWe learnedof theearlierequationthat C’ is the set of solutions of the equation 9 9 Hyperbolas from ﬂipping + = 1 The matrix D scai:s the -coordinatesThe matrixin theD plane byc a,scalesand the i-coordinates in the plane−by1 0a, and = We can ﬂip the hyperbola H over the y-axis using the matrix , ( ) = 0 1 it scales the g-coordinates by b. What’s drawn below is D(C’), (v.hich) is the it scales the y-coordinates by b. What’s drawn below is D(C’), which is the shape resulting from scaling the matrix that replaces x with −x and does not alter y. C’ horizontallyshapeby a resultingand verticallyfrom byscalingb. It’sC’anhorizontally by a and vertically by b. It’s an example of an ellipse. example of an ellipse. 0_ 0_ -0 I1c -0 0_ -0

—I

209 Using POTS. the equation for this distorted Usingcircle,POTS,the theellipseequationD(C’).foris this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix the matrix D1=( 0) ?) D1=(

187 187 The shape resulting from ﬂipping the hyperbola Hc over the y-axis is also called a hyperbola. Its equation is obtained by precomposing the equation −1 0 −1 0 for Hc, the equation xy = c, with the inverse of , which is 0 1 0 1 itself, the matrix that replaces x with −x. %____; II 0 x --‘ TT1- ,J,.J, I’ c0

So the equation for Hc after being ﬂipped over the y-axis is (−x)y = c which is equivalent to xy = −c

To summarize:

Let c > 0. The equation for Hc ﬂipped over the y-axis is xy = −c. 0_ -0 tr’

210 Example.

• The equation for H3 after being ﬂipped over the y-axis is xy = −3.

*************

Ellipses and hyperbolas are conics The equations that we’ve seen for ellipses and hyperbolas in this chapter x2 y2 are quadratic equations in two variables: a2 + b2 = 1, xy = c, and xy = −c. Thus, the solutions of these equations, the ellipses and hyperbolas above, are examples of conics.

211 Ellipses and Hyperbolas Exercises

In this chapter we’ll revisit some examplesFor #1-6,of comcs. match the numbered pictures with the lettered equations. Ellipses 1.) 4.) If you begin with the unit circle, C’, and you scale i-coordinates by some nonzero number a, and you scale y-coordinates by someEllipsesnonzeroEllipsesnumberandb,andHyperbolasHyperbolas the resulting shape in the plane is called an ellipse.(J 3 In thisInchapterthis chapterwe’ll we’llrevisitrevisitsome someexamplesexamplesof comcs.of comcs. EllipsesEllipses If youIfbeginyou beginwith thewithunitthe circle,unit circle,C’, andC’,youandscaleyou scalei-coordinatesi-coordinatesby someby some nonzerononzeronumbernumbera, anda, youandscaleyou scaley-coordinatesy-coordinatesby someby somenonzerononzeronumbernumberb, b, the resultingtheE1resultingshapeshapein theinplanethe planeis calledis calledan ellipse.an ellipse. 2.) 5.) -3 3 3 Let’s begin again with the unit circle C’, the circle of radius 1 centered at (0, 0). We learned earlier that C’ is the set of solutions of the equation 9 9 E1E1 The matrix D scales th: i-coordinates in the plane by a, and ( ) -3 -3 it scales the y-coordinates by b. What’s drawn below is D(C’). which is the Let’s Let’sbeginbeginagainagainwith thewithunitthe circleunit circleC’, theC’,circlethe circleof radiusof radius1 centered1 centeredat at shape resulting from scaling C’ horizont ally by a and vertically by b. It’s an (0, 0).(0,We0).learnedWe learnedearlierearlierthat C’thatis C’theissettheofsetsolutionsof solutionsof theofequationthe equation example of an ellipse. 3.)9 99 6.) 9 The matrixThe matrixD D( )( scales) scalesth: i-coordinatesth: i-coordinatesin theinplanethe planeby a, byanda, and it scalesit scalesthe y-coordinatesthe y-coordinatesby b. byWhat’sb. What’sdrawndrawnbelowbelowis D(C’).is D(C’).whichwhichis theis the shapeshaperesultingresultingfrom fromscalingscalingC’ horizontC’ horizontally byallya andby averticallyand verticallyby b. byIt’sb.anIt’s an exampleexampleof an ofellipse.an ellipse.

Using POTS, the equation for this distorted circle, the ellipse D(C1), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix D1=(A.)?)xy = 4 B.) x2 + 4y2 = 1 C.) x2 + y2 = 9

187 2 D.) xy = −4 E.) x2 + y = 1 F.) x2 + y2 = 1 UsingUsingPOTS,POTS,the equationthe equationfor9 thisfor16distortedthis distortedcircle,circle,the ellipsethe ellipse25D(C1),D(C1),is is obtainedobtainedby precomposingby precomposingthe equationthe equationfor for C’, the equation x2 =+ y2 = 1, by 212 C’, the equation x2 + y2 1, by the matrixthe matrix D1=(D1=(?) ?)

187 187 A For #7-10 write an equation for each of the given shapes in the plane. dotted lines are not part of the shapes. theyre just drawn to provide a point of reference.)

7.) 9.) (f)

For #7-10, write an equation for each of the given shapes in the plane. Your 5 answers should have the form (y − q) = a(x − p), (x − p)2 + (y − q)2 = r2, 2 2 (x − p)(y − q) = c, or (x−p) + (y−q) = 1. (As with the x- and y-axes, the For #7-10 write an equationa2 for beach2 of the given shapes in the plane. A For #7-10 write an equation for each of the given shapes in the plane. A dotteddotted lineslines are notare partnot part of theof the shapes.shapes. They’retheyre justjust drawndrawn toto provideprovide a point ofdotted reference.)lines are not part of the shapes. theyre just drawn to provide a pointpointof reference.)of reference.)

7.) 9.) 8.) 10.)7.) 7.) 9.) 9.) (f) (f)(35:

5 5

A For #7-10 write an equation for each of the given shapes in the plane. dotted lines are not part of the shapes. theyre just drawn to provide a point of reference.)

8.)8.)10.) 10.) 8.) 10.) 7.) 9.) (f) For #11-14, multiply the matrices. (35:(35:

1) 3 7 11.) 13.) 5 ( —3 (a ) ( 9

(2 0 0 “3 —9 (2 1 19) ( 14.) -. 0 3)\O 0 1 8.) 10.) 194 For multiply the matrices. ForFor#11-14,#11-14, #11-14,multiply multiplythe matrices. the matrices. (35:

1 1)−1 03 3 1 07 5 7 11.)11.) 1) 3 13.) 13.) 7 11.) —3 13.) −1( 1 (—32 −3 (a (a))(0(9 1 9 8 9

1 —9 (2 0 0 “3—9 (21 19)(2 02 0 (0 2 0 14.)“3 3(2−12 2 1 19)12.)-. 0 3)\O( 1 14.) 14.) 0 1 -. 0 3)\O0 3 0 3 0 1 01 0 1

194 194 213 For #11-14, multiply the matrices.

1) 3 7 11.) 13.) ( —3 (a ) ( 9

(2 0 0 “3 —9 (2 1 19) ( 14.) -. 0 3)\O 0 1

194