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Chapter 4 Ellipses and Hyperbolas

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Chapter 4 Ellipses and Hyperbolas Chapter 4 Ellipses and Hyperbolas 4.1 Ellipses Ellipse with two given foci 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 1 x2 y2 + =1 with b2 := a2 − c2. (4.1) a2 b2 b P a c a 2 2 O F − a A O F A a F c F c 1 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; (a2 − c2)x2 + a2y2 = a2(a2 − c2). Writing b2 := a2 − c2,wehaveb2x2 + a2y2 = a2b2. 212 Ellipses and Hyperbolas The directrices and eccentricity of an ellipse 2 =( ) x2 + y =1 2 If P x, y is a point on the ellipse a2 b2 , its square distance from the focus F is 2 | | = c − a PF a x c . P 2 2 − a A O F A a c F 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. 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 2 2 2 2 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 . 4.2 Tangents and normals of an ellipse 213 4.2 Tangents and normals of an ellipse Construction of tangent at P x1x y1y =( 1 1) + =1 If P x ,y , 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 circlex +y = a at Q 2 a2 1 + 2 = = 0 is the line x x y y a . Both tangents intersect the x-axis at T 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 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 214 Ellipses and Hyperbolas Exercise 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 a F O F 2. Prove the following reflection property of the ellipse. P 2 2 a A O F A a c F c 4.2.1 Evolute of an ellipse 2 x2 + y =1 Consider the ellipse a2 b2 with parametrization x = a cos θ, y = b sin θ, wherea>b(so that the foci are on the x-axis). The center of curvature at P = P (θ) is the a2−b2 · cos3 − a2−b2 · sin3 point a θ, b θ . It can be constructed as follows. (1) Let the tangent at P intersect the axes at X and Y . (2) Complete the rectangle OXQY . (3) Construct the perpendicular from O to the line PQ, to intersect the normal at K. The point K is the center of curvature at P . 4.2 Tangents and normals of an ellipse 215 Q Y P O X K The evolute is the parametric curve a2 − b2 a2 − b2 x = · cos3 θ, y = − · sin3 θ. a b Eliminating t,wehave 2 2 4 (ax) 3 +(by) 3 = c 3 . This curve is called an astroid. P O Q 216 Ellipses and Hyperbolas 4.3 Hyperbolas Hyperbola with two given foci Given two points F and F in a plane, the locus of point P for which the distances PF and PF have a constant difference is a hyperbola with foci F and F . b P c O a2 a F F c Assume FF =2c and the constant difference |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 hyperbola if and only if | (x + c)2 + y2 − (x − c)2 + y2| =2a. This can be reorganized as x2 y2 − =1 with b2 := c2 − a2. (4.2) a2 b2 The directrices and eccentricity of a hyperbola 2 x2 y If P =(x, y) is a point on the hyperbola 2 − 2 =1, its square distance from the focus a b 2 3 | | = c − a := c = · − a F is PF a x c . Writing e a ,wehavePF e x e . The hyperbola can = a also be regarded as the locus of point P whose distances from F (focus) and the line x e 2 1 | | = + a (directrix) bear a constant ratio e> (the eccentricity). Similarly, PF e x c . The =(− 0) = − a point F ae, and the line x e form another pair of focus and directrix of the same hyperbola. There is an easy parametrization of the hyperbola: P (θ)=(a sec θ, b tan θ). Given a point T on the auxiliary circle C (O, a), construct (i) the tangent to the circle at T to intersect the x-axis at A, (ii) the line x = b to intersect the line OT at B, (iii) the “vertical” line at A and the “horizontal” line at B to intersect at P . The point P has coordinates (a sec θ, b tan θ), and is a point on the hyperbola H . If Q is the pedal of T on the x-axis, then the line PQis tangent to the hyperbola. 2 2 2 3 2 2 2 2 2 x b 2 2 2 c 2 2 PF =(x − c) + y =(x − c) + b 1 − 2 = 1 − 2 x − 2cx + c + b = 2 x − 2cx + a = a a a c 2 a x − a . 4.4 Tangents and normals of a hyperbola 217 B P T θ −c O Q a b c A 4.4 Tangents and normals of a hyperbola 2 H : x2 − y =1 Let P be a point on the hyperbola a2 b2 . The circles with diameters PF1 and PF2 are tangent to the auxiliary circle C (O, a). The line joining the points of tangency is tangent to H at P . P a F O F Exercise 1. If P and Q are on one branch of a hyperbola with foci F and G, show that F and G are on one branch of a hyperbola with foci P and Q. 2. Let P be a point on a hyperbola. The intersections of the tangent at P with the 218 Ellipses and Hyperbolas asymptotes, and the intersection of the normal at P with the axes lie on a circle through the four points passes through the center of the hyperbola. P a F O F a2+b2 sec a2+b2 tan The center of the circle is 2a θ, 2b θ . The locus is the hyperbola 4(a2x2 − b2y2)=(a2 + b2)2. 4.4.1 Evolute of a hyperbola 2 x2 − y =1 ( )=( sec tan ) For the hyperbola a2 b2 with parametrization x, y a t, b t , the evolute is the parametric curve a2 + b2 a2 + b2 x = · sec3 t, y = − · tan3 t. a b P O Q 4.5 Rectangular hyperbolas 219 4.5 Rectangular hyperbolas 2 Example 4.1. Four points on the rectangular hyperbola xy = c with parameters t1, t2, t3, t4 form an orthocentric system if and only if t1t2t3t4 = −1. Proof. Consider three points on the hyperbola with parameters t1, t2, t3. The perpendicu- lars from each one of them to the line joining the other two are the lines 2 t1t2t3x − t1y + c(1 − t1t2t3)= 0, 2 t1t2t3x − t2y + c(1 − t1t2t3)= 0, 2 t1t2t3x − t3y + c(1 − t1t2t3)= 0. c 4 1 2 3 4 = −1 ( )= 4 If t is such that t t t t , then it is easy to see that x, y ct , t4 satisfies each of the above equations.
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