Problem #1: The polar equation (7 cos²(0)+5)(A) a hyperbola (B) a parabola (C) a circle with center (1, 0) and radius 9(D) a circle with center (0, 1) and radius 9 (E) an ellipse (F) a straight lineSelectProblem #1:represents,Save

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Answered: roblem #1: The polar equation ²(7…

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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7. Consider the ellipse = 1, 0 < a < b. (a) Write down a set of parametric equations that describe the entire ellipse. Be sure to state the domain of your parametric equations. (Hint: Your parametric equations should involve trigonometric functions.) (b) The curvature k of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature k = 0 at every point. For a curve C with parametric equations r = r(t), y = y(t), then at the point (r(t), y(t)) the curvature can be calculated by |(t)y"(t) – y'(t)r" (t)|| (r'(t))² + (y(t))?)³/2 K= Use your parametric equations from (a) to calculate the curvature of the ellipse at (2(t), y(t)). (c) Use your answer to (b) to find the point(s) on the curve at which the curvature is maxi- mized. Justify your answer with calculations. (d) Write 1-2 sentences explaining whether your answer to (c) makes sense based on the illustration of the ellipse aboe.
Prove that the graph of the equation Ax2 + Cy2 + Dx + Ey + F = 0 is one of the following (except in degenerate cases).
Please do the following handwritten
8. Use the following conic sections to answer: x2 – 2x + 2y2 – 8y + 7 = 0 a. Graph the conic section b. Give the key features of the conic section (e.g. center, vertex, asymptotes, eccentricity, etc.) c. Transform the conic section into polar. Make sure you simplify to r=equation.
(y Find the foci of the following hyperbola: + 1)² _ (x −1)² =1 25 85°F Clear Esc ○ (−1+√34,-1); (−1 - √34, −1) O (1,4); (1, -2) O (6, -1); (-4, -1) ○ (1,−1+√34); (1, −1 – √34) F1 ! 1 F2 @ 2 F3 #M 3 F4 LA $ 4 F5 Q Search 20 vr do % F6 凶 F7 6 F8 & 7
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.
7. Consider the ellipse = 1, 0 < a < 6. 62 (a) Write down a set of parametric equations that describe the entire ellipse. Be sure to state the domain of your parametric equations. (Hint: Your parametric equations should involve trigonometric functions.) (b) The cutvature K of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature k = 0 at every point. For a curve C with parametric equations x = (I(t), y(t)) the curvature can be calculated by x(t), y = y(t), then at the point f'(t)y"(t) – ' (t)æ"(t)| (r'(t))2 + (())?)3/2 - Use your parametric equations from (a) to calculate the curvature of the ellipse at (2(t), y(t)). (c) Use your answer to (b) to find the point(s) on the curve at which the curvature is maxi- mized. Justify your answer with calculations. (d) Write 1-2 sentences explaining whether your answer to (c) makes sense based on the illustration of the ellipse above. 4.

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roblem #1: The polar equation ²(7 cos²(0)+5) (A) a hyperbola (B) a parabola (C) a circle with center (1, 0) and radius 9 (D) a circle with center (0, 1) and radius 9 (E) an ellipse (F) a straight line Problem #1: represents, Colect