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 $\left(0,2\right)$, is clockwise, and requires 2 seconds for a complete revolution.

Short answer

The parametric equation is$x=2\cos \left(\mathrm{\pi t}\right),y=-3\sin \left(\mathrm{\pi t}\right),0\le t\le 2$

Step-by-step solution

Step 1. Given Information

We are given an ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$

The graph of ellipse is shown as

Step 2. Finding Parametric Equation

The parametric equation is $x=a\cos \left(\omega t\right),y=b\sin \left(\omega t\right)$

$x=2\cos \left(\omega t\right),y=-3\sin \left(\omega t\right)$

Since one revolutions requires 2 seconds, the period is $$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega }=2 \\ 2\omega =2\mathrm{\pi } \\ \mathrm{\omega }=\mathrm{\pi } \end{gathered}$$

As a result,$x=2\cos \left(\mathrm{\pi }t\right),y=-3\sin \left(\mathrm{\pi }t\right)$