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(a) The curve C is defined by the equation 4x²9y²8x18y +31 = 0. What type of conic section is C? Find (where these exist) its centre, eccentricity, foci, asymptotes and axis intercepts. Sketch C.

(a) The curve C is defined by the equation 4x²9y²8x18y +31 = 0. What type of conic section is C? Find (where these exist) its centre, eccentricity, foci, asymptotes and axis intercepts. Sketch C.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. (a) The curve C is defined by the equation 4x² 9y²8x 18y + 31 = 0. What type of conic section is C? Find (where these exist) its centre, eccentricity, foci, asymptotes and axis intercepts. Sketch C. (b) A ellipse C has equation = x² + q² 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 = b sint. (You may assume facts about the standard parametrisation of a circle, so long as you state them clearly.) (a cos q, b sin q) both lie on C. Prove (ii) Points P = (a cos p, b sin p) and Q that the chord PQ has gradient b + (P + a) (You may assume any of the following identities: cos - a sin sin = 2 sin cot cos+cos o = 2 cos = 1 (0 + 0) (070) COS 2 2 (₁ + $) Cos (²₂0) COS 2 2 (*2*)sin(z). (iii) Hence show that the tangent to C at the point P has equation - cos=-2 sin sin p b (Hint: take the limit at q gets closer to p.) (cosp) x + a y = 1.
Transcribed Image Text:3. (a) The curve C is defined by the equation 4x² 9y²8x 18y + 31 = 0. What type of conic section is C? Find (where these exist) its centre, eccentricity, foci, asymptotes and axis intercepts. Sketch C. (b) A ellipse C has equation = x² + q² 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 = b sint. (You may assume facts about the standard parametrisation of a circle, so long as you state them clearly.) (a cos q, b sin q) both lie on C. Prove (ii) Points P = (a cos p, b sin p) and Q that the chord PQ has gradient b + (P + a) (You may assume any of the following identities: cos - a sin sin = 2 sin cot cos+cos o = 2 cos = 1 (0 + 0) (070) COS 2 2 (₁ + $) Cos (²₂0) COS 2 2 (*2*)sin(z). (iii) Hence show that the tangent to C at the point P has equation - cos=-2 sin sin p b (Hint: take the limit at q gets closer to p.) (cosp) x + a y = 1.
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