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Advanced Algebra and Geometry 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 .
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