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 $\left(0,3\right)$, is counterclockwise, and requires 1 second for a complete revolution.

Short answer

The parametric equation is$x=-2\cos \left(2\mathrm{\pi t}\right),y=3\sin \left(2\mathrm{\pi t}\right),0\le t\le 1$

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 in the figure

Step 2. Finding Parametric Equations

The ellipse is rotated at $\left(0,3\right)$and shown as

we can use parametric equations $x=a\cos \left(\omega t\right),y=b\sin \left(\omega t\right)$.

The ellipse is rotated towards $\left(-2,0\right)$.So, $a=-2,b=3$

Since one revolution requires 1 seconds, the period is $$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega }=1 \\ \omega =2\mathrm{\pi } \end{gathered}$$

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