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Parametric equations for an ellipse Consider the
Chapter 9, Problem 99AE(choose chapter or problem)
Parametric equations for an ellipse Consider the parametric equations
x = a cos t + b sin t, y = c cos t + d sin t,
where a, b, c, and d are real numbers
a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K\), where A, B, C, and K are constants
b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the x- and y-axes provided ab + cd = 0.
c. Show that the equations describe a circle provided ab + cd = 0 and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\).
Questions & Answers
QUESTION:
Parametric equations for an ellipse Consider the parametric equations
x = a cos t + b sin t, y = c cos t + d sin t,
where a, b, c, and d are real numbers
a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K\), where A, B, C, and K are constants
b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the x- and y-axes provided ab + cd = 0.
c. Show that the equations describe a circle provided ab + cd = 0 and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\).
ANSWER:Solution 99AE
a. Now squaring and adding the equations (3) and (4) we get
(dx - by) 2 + (cx - ay)2 = (ad - bc)2
x2 (c2 +d2) + y2 (a2 + b2) -2xy (ac +bd) = (ad - bc) 2
This is in the form Ax2 + Cy2 +Bxy = k
Here B2 - 4AC = 4 (ac + bd)2 -4(c2