Question

Find parametric equations for an object that moves along the ellipse$\frac{x^2}{16}+\frac{y^2}{9}=1$

where the motion begins at $\left(4,0\right)$, is counterclockwise, and requires 4 seconds for a complete revolution.

Short answer

The parametric equation is:

$$\begin{gathered} x=4\cos \left(\frac{\mathrm{\pi }}{2}t\right) \\ y=3\sin \left(\frac{\mathrm{\pi }}{2}t\right) \end{gathered}$$

Step-by-step solution

Step 1. Given information:

The equation of ellipse:

$\frac{x^2}{16}+\frac{y^2}{9}=1$

Time taken by object to complete one cycle is 4 seconds.

Step 2. Draw the graph of the ellipse using a graphing calculator.

The graph of the ellipse is:

Step 3. To find the parametric equation of the ellipse.

Let,

$$\begin{gathered} x=4\cos \left(\omega t\right) \\ y=3\sin \left(\omega t\right) \end{gathered}$$, where $\omega$is some constant.

These parametric equations satisfy the equation of the ellipse.

Since,

$$\begin{gathered} \frac{{\left(4\cos \left(\omega t\right)\right)}^2}{16}+\frac{{\left(\sin \left(\omega t\right)\right)}^2}{9} \\ =\frac{16{\cos }^2\left(\omega t\right)}{16}+\frac{9{\sin }^2\left(\omega t\right)}{9} \\ ={\cos }^2\left(\omega t\right)+{\sin }^2\left(\omega t\right) \\ =1 \end{gathered}$$

Furthermore, when $t=0$, $x=4$and when $y=0$.

For the motion to be counterclockwise, the motion has to begin with the value of $x$decreasing and $y$increasing as $t$increases. This requires that $\omega >0$.

Finally since 1 revolution requires 4 seconds, the period is $$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{4} \\ \omega =\frac{\mathrm{\pi }}{2} \end{gathered}$$.

The parametric equations that satisfy the conditions stipulated are :

$$\begin{gathered} x=4\cos \left(\frac{\mathrm{\pi }}{2}t\right) \\ y=3\sin \left(\frac{\mathrm{\pi }}{2}t\right) \end{gathered}$$