Question

In Problems 39 – 42, find parametric equations for an object that moves along the ellipse$\frac{x^2}{4}+\frac{y^2}{9}=1$with the motion described.

The motion begins at $\left(2,0\right)$, is counterclockwise, and requires 3 seconds for a complete revolution.

Short answer

The parametric equations are $x=2\cos \left(\frac{2\mathrm{\pi }}{3}t\right),y=3\sin \left(\frac{2\mathrm{\pi }}{3}t\right),0\le t\le 3$

Step-by-step solution

Step 1. Given Information

The given ellipse is $\frac{x^2}{4}+\frac{y^2}{9}=1$

The graph is shown as

Step 2. Finding Parametric Equations

The motion begins at$\left(2,0\right)$ is counterclockwise,

We can use $x=a\cos \left(\omega t\right),y=b\sin \left(\omega t\right)$

Since one revolutions requires 3 seconds

$T=\frac{2\mathrm{\pi }}{\omega }=3$

$\omega =\frac{2\mathrm{\pi }}{3}$

Thus,$$\begin{gathered} x=a\cos \left(\omega t\right),y=b\sin \left(\omega t\right) \\ x=2\cos \left(\frac{2\mathrm{\pi }}{3}t\right),y=3\sin \left(\frac{2\mathrm{\pi }}{3}t\right) \end{gathered}$$