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 .