Question

(1 pt) Find a parametrization of the ellipse centered at the origin in the xy-plane that has major diameter 14 along the x-axis, minor diameter 10 along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point (7,0) correspond to t=0. Use t as the parameter for all of your answers.

Answer 1

`x=7\cos t, y=5\sin t\text{ for }0\le t<2\pi`

Explanation:

The parametrization of an ellipse center at origin and counter-clockwise is given by the formulas:

`x=a\cos t, y=b\sin t\text{ for }0\le t<2\pi`

Where “a” is the radius of the major axis along the x-axis and “b” is the radius of the minor axis along the y-axis. Since the major diameter along the x-axis is 14 then its radius is its half, thus a=7. Similarly, since the minor diameter along the y-axis is 10, then its radius is half of it, thus b=5

Therefore, the parametric equations for the ellipse become:

`x=7\cos t, y=5\sin t\text{ for }0\le t<2\pi`

Notice at t=0 we get:

`x=7\cos(0) =7, y=5\sin(0)=0`

which satisfies that the parametrization at t=0 makes the point (7,0)