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(b) A ellipse C has equation a² b where a b>0. (i) Show that the point P = (x, y) lies on C if and only if there exists some t = [0, 2π) such that x = a cost and y = bsint. (You may assume facts about the standard parametrisation of a circle, so long as you state them clearly.) (ii) Points P = (a cos p, b sin p) and Q = (a cos q, b sin q) both lie on C. Prove that the chord PQ has gradient 62 p+q +(²+²) a 2 (You may assume any of the following identities: cot sin sin = 2 sin cos + coso = 2 cos 1 COS 2 2 (ª + º) • (070) (0+0) cos (²70) COS 2 2 0 (2*)sin(*z*). (iii) Hence show that the tangent to C at the point P has equation os p (COSP) x + (Simp) a (Hint: take the limit at q gets closer to p.) cos cos=-2 sin 0 + y = 1.

(b) A ellipse C has equation a² b where a b>0. (i) Show that the point P = (x, y) lies on C if and only if there exists some t = [0, 2π) such that x = a cost and y = bsint. (You may assume facts about the standard parametrisation of a circle, so long as you state them clearly.) (ii) Points P = (a cos p, b sin p) and Q = (a cos q, b sin q) both lie on C. Prove that the chord PQ has gradient 62 p+q +(²+²) a 2 (You may assume any of the following identities: cot sin sin = 2 sin cos + coso = 2 cos 1 COS 2 2 (ª + º) • (070) (0+0) cos (²70) COS 2 2 0 (2*)sin(*z*). (iii) Hence show that the tangent to C at the point P has equation os p (COSP) x + (Simp) a (Hint: take the limit at q gets closer to p.) cos cos=-2 sin 0 + y = 1.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 18T
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(b) A ellipse C has equation x² + a² 6² where a b>0. (i) Show that the point P = (x, y) lies on C if and only if there exists some t = [0, 2π) such that x = a cost and y = bsint. (You may assume facts about the standard parametrisation of a circle, so long as you state them clearly.) (ii) Points P = (a cosp, b sin p) and Q = (a cos q, b sin q) both lie on C. Prove that the chord PQ has gradient b p+q a 2 (You may assume any of the following identities: sin sin = 2 sin cot - cos+coso = 2 cos 0 +6 2 0 (6 + 6) sin ( 02 2 (iii) Hence show that the tangent to C at the point P has equation sin p |y=1. b (Hint: take the limit at q gets closer to p.) 1 (cosp) a (0+) cos (07) 2 2 cos cos=-2 sin (² x + COS (³₂0) 2
Transcribed Image Text:(b) A ellipse C has equation x² + a² 6² where a b>0. (i) Show that the point P = (x, y) lies on C if and only if there exists some t = [0, 2π) such that x = a cost and y = bsint. (You may assume facts about the standard parametrisation of a circle, so long as you state them clearly.) (ii) Points P = (a cosp, b sin p) and Q = (a cos q, b sin q) both lie on C. Prove that the chord PQ has gradient b p+q a 2 (You may assume any of the following identities: sin sin = 2 sin cot - cos+coso = 2 cos 0 +6 2 0 (6 + 6) sin ( 02 2 (iii) Hence show that the tangent to C at the point P has equation sin p |y=1. b (Hint: take the limit at q gets closer to p.) 1 (cosp) a (0+) cos (07) 2 2 cos cos=-2 sin (² x + COS (³₂0) 2
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