Advanced Algebra and Geometry
Paul Yiu
Department of Mathematics Florida Atlantic University
Fall 2016
November 14, 2016
Contents
12 Conics 301 12.1 Parabolas ...... 301 12.1.1 Chords and tangents ...... 302 12.1.2 Triangle bounded by three tangents ...... 304 12.2 Ellipses ...... 306 12.2.1 The directrices and eccentricity of an ellipse ...... 306 12.2.2 The auxiliary circle and eccentric angle ...... 307 12.2.3 Tangents of an ellipse ...... 307 12.2.4 Ellipse inscribed in a triangle ...... 308 12.3 Hyperbolas ...... 311 12.3.1 The directrices and eccentricity of a hyperbola ...... 311 12.3.2 Tangents of a hyperbola ...... 312 12.3.3 Rectangular hyperbolas ...... 313 12.3.4 A theorem on the tangents from a point to a conic ...... 314
13 General conics 317 13.1 Classification of conics ...... 317 13.2 Pole and polar ...... 318 13.3 Condition of tangency ...... 319 13.4 Parametrized conics ...... 320 13.5 Rectangular hyperbolas and the nine-point circle ...... 321
14 Conic solution of the (O, H, I) problem 325 14.1 Ruler and compass construction of circumcircle and incircle ...... 325 14.2 The general case: intersections with a conic ...... 325 14.3 The case OI = IH ...... 328
Chapter 12
Conics
12.1 Parabolas
Given a point F (focus) and a line L (directrix), the locus of a point P which is equidistant from F and L is a parabola P. Let the distance between F and L be 2a. Set up a Cartesian coordinate system such that F =(a, 0) and L has equation x = −a. The origin O is clearly on the locus. Its distances from F and L are both a.
L
Q P (x, y)
directrix
− a O F =(a, 0)
Let P (x, y) be a point equidistant from F =(a, 0) and L . Then (x − a)2 + y2 = (x + a)2. Rearranging terms, we have
y2 =4ax.
The parabola P can also be described parametrically: x = at2, y =2at.
We shall refer to the point P (t):=(at2, 2at) as the point on P with parameter t.Itisthe intersection of the line y =2at and the perpendicular bisector of FQ, Q =(−a, 2at) on the directrix. 302 Conics
12.1.1 Chords and tangents
The line joining the points P (t1) and P (t2) has equation
2x − (t1 + t2)y +2at1t2 =0.
Letting t1,t2 → t, we obtain the equation of the tangent at P (t):
x − ty + at2 =0. ( ) 1 (− 2 0) The tangent at P t has slope t and intersects the axis of the parabola at at , .It can be constructed as the perpendicular bisector of FQ, where Q is the pedal of P on the directrix.
Exercise
1. (Focal chord) The line joining two distinct points P (t1) and P (t2) on P passes through the focus F if and only if t1t2 +1=0.
2. (Intersection of two tangents) Show that the tangents at the points t1 and t2 on the 2 parabola y =4ax intersect at the point (at1t2,a(t1 + t2)).
3. Justify the construction suggested by the diagram below for the construction of the tangent at a point P on a parabola.
L
P
O F
4. (Tangents from a point to a parabola) Given a point P , construct the circle with diameter PF and let it intersect the tangent at the vertex at Q1 and Q2. Then the lines PQ1 and PQ2 are the tangents from P to the parabola.
5. Let P1P2 be a focal chord of a parabola P with midpoint M. The tangents at P1 and P2 intersect at Q. Show that (i) Q lies on the directrix, (ii) these tangents are perpendicular to each other, and (iii) QM is parallel to the axis of the parabola, and (iv) QM and intersects P at its own midpoint. 12.1 Parabolas 303
T1
Q1
P
O F
Q2 T2
6. Justify the following construction of the chord of a parabola which has a given point M as its midpoint: Let Q be the pedal of M on the directrix. Construct the circle M(Q) to intersect the perpendicular from M to QF . These two intersections are on the parabola, and their midpoint is M.
L P1
Q M
O F
P2
7. A circle on any focal chord of a parabola as diameter cuts the curve again at two points P and Q. Show that as the focal chord varies, the line PQ passes through a fixed point. 8. (Chords orthogonal at the vertex) Let PQ be a chord of a parabola with vertex O such that angle POQis a right angle. Find the locus of the midpoint of PQ. 9. Find the locus of the point whose two tangents to the parabola y2 =4ax make a given angle α. 10. PQ is a focal chord of a parabola with focus F . Construct the circles through F tangent to the parabola at P and Q. What is the locus of the second intersection of the circles (apart from F )? 304 Conics
12.1.2 Triangle bounded by three tangents
2 Proposition 12.1. The points P (t1), P (t2), P (t3), P (t4) on the parabola y =4ax are concyclic if and only if t1 + t2 + t3 + t4 =0.
Theorem 12.2. The circumcircle of the triangle bounded by three tangents to a parabola passes through the focus of the parabola.
L
P1
A3
L3 P2
A2 F
A1
L1
P3 L2
=1 2 3 = − + 2 =0 Proof. For i , , , the tangent at ti is the line Li x tiy ati . We find λ1, λ2, λ3 such that
λ1L2 · L3 + λ2L3 · L1 + λ3L1 · L2 =0 represents a circle. This requires the coefficients of x2 and y2 to be equal, and that of xy equal to 0:
(1 − t2t3)λ1 +(1− t3t1)λ2 +(1− t1t2)λ3 =0, (t2 + t3)λ1 +(t3 + t1)λ2 +(t1 + t2)λ3 =0.
Solving these equations, we have 1 − t3t1 1 − t1t2 1 − t1t2 1 − t2t3 1 − t2t3 1 − t3t1 λ1 : λ2 : λ3 = : : t3 + t1 t1 + t2 t1 + t2 t2 + t3 t2 + t3 t3 + t1 2 2 2 =(t2 − t3)(1 + t1):(t3 − t1)(1 + t2):(t1 − t2)(1 + t3).
2 2 With these values of λ1, λ2, λ3, we compute the common coefficient of x and y as 2 (t2 − t3)(1 + t1)=−(t2 − t3)(t3 − t1)(t1 − t2). 12.1 Parabolas 305
Also,
coefficient of x 2 2 2 = (t2 − t3)(1 + t1) · a(t2 + t3)
= a(t2 − t3)(t3 − t1)(t1 − t2)(1 + t2t3 + t3t1 + t1t2), coefficient of y 2 2 2 = − (t2 − t3)(1 + t1) · a(t2 + t3)
= a(t2 − t3)(t3 − t1)(t1 − t2)(t1 + t2 + t3 − t1t2t3), the constant term 2 2 2 2 = (t2 − t3)(1 + t1) · a t2t3 2 = − a (t2 − t3)(t3 − t1)(t1 − t2)(t2t3 + t3t1 + t1t2).
Cancelling a common factor −(t2 − t3)(t3 − t1)(t1 − t2), we obtain the equation of the circle as 2 2 2 x + y − a(1 + σ2)x − a(σ1 − σ3)y + a σ2 =0, where σ1, σ2, σ3 are the elementary symmetric functions of t1, t2, t3. This circle clearly passes through the focus F =(a, 0).
Exercise 1. Prove that the orthocenter of the triangle formed by three tangents to a parabola lies on the directrix.
2. Show that the parabola tangent to the internal and external bisectors of angle B and C of triangle ABC has focus at vertex A and directrix the line BC. 1
Exercise 1. (Parabolas with a common vertex and perpendicular axes) The two parabolas P : y2 =4ax and P : x2 =4by have a common vertex O and perpendicular axes. Let A be their common point other than O. The tangent at A to P intersects P at B and the tangent at A to P intersects P at C. Show that the line BC is a common tangent of the parabolas.
2. (Conformal focal parabolas) Two parabolas have a common focus F ; their directrices intersects at a point A. Show that the perpendicular bisector of AF is the common tangent of the two parabolas.
1 Solution. Let I, Ib, Ic be the incenter and the B-, C-excenters of triangle ABC. The two bisectors of angle B and the internal bisector of angle C bound the triangle IBIc. The two bisectors of angle C and the internal bisector of angle B bound the triangle ICIb. Apart from I, the circumcircles of these triangles intersect at A. This is the focus of the parabola. The reflections of A in the bisectors of angles B and C are points on the sideline BC. Therefore, the line BC is the directrix. 306 Conics
3. (a) Find the condition for the two parabolas y2 =4ax and y2 =4b(x − c) to have a common tangent. (b) Construct the common tangent of the two parabolas.
12.2 Ellipses
Given two points F and F in a plane, the locus of point P for which the distances PF and PF have a constant sum is an ellipse with foci F and F . Assume FF =2c and the constant sum PF+PF =2a for a>c. Set up a coordinate system such that F =(c, 0) and F =(−c, 0). A point (x, y) is on the ellipse if and only if (x − c)2 + y2 + (x + c)2 + y2 =2a.
This can be reorganized as 2
x2 y2 + =1 with b2 := a2 − c2. (12.1) a2 b2
b P a
c a 2 2 O F − a A O F A a F c F c
12.2.1 The directrices and eccentricity of an ellipse
2 =( ) x2 + y =1 3 If P x, y is a point on the ellipse a2 b2 , its square distance from the focus F is 2 |PF| = c x − a . a c := c = · − a Writing e a ,wehavePF e x e . The ellipse can also be regarded as the locus = a of point P whose distances from F (focus) and the line x e (directrix) bear a constant 2 | | = + a =(− 0) ratio e (the eccentricity). Similarly, PF e x c . The point F ae, and the = − a line x e form another pair of focus and directrix of the same ellipse. 2 2 2 2 2 From (x + c) + y =2a − (x − c) + y ,wehave 2 2 2 2 2 2 2 (x + c) + y =4a +(x − c) + y − 4a (x − c) + y ; 4a (x − c)2 + y2 =4a2 − 4cx; a2((x − c)2 + y2)=(a2 − cx)2; 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (a − c )x + a y = a (a − c ). Writing b := a − c ,wehave b x + a y = a b . 2 2 2 3 2 2 2 2 2 x b 2 2 2 c 2 2 PF =(x − c) + y =(x − c) + b 2 − 1 = 1+ 2 x − 2cx + c − b = 2 x − 2cx + a = a a a c 2 a x − a . 12.2 Ellipses 307
P
2 2 − a A O F A a c F c
12.2.2 The auxiliary circle and eccentric angle The circle with the major axis as diameter is called the auxiliary circle of the ellipse. Let P be a point on the ellipse. If the perpendicular from P to the major axis intersects the auxiliary circle at Q (on the same side of the major axis), we write Q =(a cos θ, a sin θ) and call θ the eccentric angle of P . In terms of θ, the coordinates of P are (a cos θ, b sin θ). This is a useful parametrization of the ellipse.
Q b P Q
−a θ a O −c O c
−b
12.2.3 Tangents of an ellipse Construction of tangent at P
=( ) x1x + y1y =1 If P x1,y1 , the tangent to the ellipse at P is the line a2 b2 . The corresponding 2 2 2 point Q on the auxiliary circle is Q =(x1,y2). The tangent to the circle x + y = a at Q 2 a2 x1x + y2y = a x T = , 0 is the line . Both tangents intersect the -axis at x1 . Therefore, from the tangent of the circle (which is perperpendicular to the radius OQ), we have the point T . Then, PT is the tangent to the ellipse at P .
Q b P
−a θ −c O c a T
−b 308 Conics
Remarks. (1) This construction can be reversed to locate the point of tangency on a line which is known to be tangent to an ellipse. (2) In terms of the eccentric angle of P , these tangents are cos θ · x +sinθ · y =a, (circle) cos θ sin θ · x + · y =1, (ellipse). a b
Construction of tangents from an external point
P Q
F O F Q
1. The circles with diameters PF and PF are tangent to the auxiliary circle at two points on the tangent to the ellipse at P .
b P P
a F O F 2 A O F A a c F c
2. Prove the following reflection property of the ellipse.
12.2.4 Ellipse inscribed in a triangle Theorem 12.3. If a, b, c are the three complex roots of a cubic polynomial f(z), then the two roots of its derivative f (z) are the foci of the ellipse which is tangent to the sides of the triangle with vertices a, b, c at their midpoints. 12.2 Ellipses 309
α
Ω γ β
G Ω
γ β α
Proof. We may assume α + β + γ =0so that the centroid is G =0and the cubic has the form f(z)=z3 + pz + q. Then, its three roots are
α = v + w, β = ωv + ω2w, γ = ω2v + ωw, for two complex numbers v and w, and ω is a primitive cube root of unity. Note that
p = αβ + γα + βγ = −(v + w)2 + v2 + w2 − vw = −3vw, q = − αβγ =(v + w)(v2 + w2 − vw)=v3 + w3.
This means f(z)=z3 − 3vw · z +(v3 + w3) and f (z)=3(z2 − vw). The two roots of f (z) are therefore the square roots of vw: √ √ Ω= vw, Ω = − vw.