SOLUTION: Find a set of parametric equations for the conic section or line below; Ellipse: Center: (5,-3); Vertices: 6 units above and below the center; Endpoint of minor Axis: 2 units left

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Find a set of parametric equations for the conic section or line below; Ellipse: Center: (5,-3); Vertices: 6 units above and below the center; Endpoint of minor Axis: 2 units left      Log On


   

Question 1143991: Find a set of parametric equations for the conic section or line below;
Ellipse: Center: (5,-3); Vertices: 6 units above and below the center; Endpoint of minor Axis: 2 units left and right of center.
A. x=5+6cos t, y=-3+2 sin t
B. x=5-2cos t, y=-3-6 sin t
C. x=5+2cos t, y=-3+6 sin t
D. x=2+5cos t, y=6-3 sin t
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let's start with a generic set of parametric equations and see how it relates to the equation of an ellipse in rectangular coordinates.

x+=+a%2Bb%2Acos%28t%29
y+=+c%2Bd%2Asin%28t%29

The parameter is t; eliminate it using sin^2+cos^2=1.

cos%28t%29+=+%28x-a%29%2Fb
sint%28t%29+=+%28y-c%29%2Fd

sin%28t%29%5E2%2Bcos%28t%29%5E2+=+%28x-a%29%5E2%2Fb%5E2%2B%28y-c%29%5E2%2Fd%5E2+=+1

The center is (a,c); the semi-major and semi-minor axes are (in some order) b and d.

Now use that analysis for the given ellipse.

The center is (a,c) = (5,-3), so the constants in the parametric equations for x and y must be 5 and -3, respectively. That eliminates answer choice D.

The semi-major axis is 6, in the y direction, so d is 6. That makes answer choice C the only possible correct answer.

It can be seen that answer choice C is in fact correct, because it shows the semi-minor axis having length b=2.

ANSWER: C. x=5+2cos t, y=-3+6 sin t