Chapter 1.7, Problem 39E

To determine

To find: The parametric equations for the curve that consists of all the possible position of the point $P$ by the angle $\theta$ as the parameter.

Answer to Problem 39E

The parametric equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

Explanation of Solution

Given:

The given figure is shown in Figure 11

  

Figure 1

Calculation:

Consider the radius of the small circle is $b$ and the radius of the large circle is $a>b$ .

From the given figure,

  $\begin{array}{l}\left|OQ\right|=a\\ \left|ON\right|=b\end{array}$

Then,

  $\left|NQ\right|=a-b$

The value of angle $QOM$ is,

  $QOM=\theta$

Then, by the property of the similar triangle angle is,

  $\left|OM\right|=a\cos \theta$

Then,

  $x=a\cos \theta$ …… (1)

Then, from the angle $QNP$ is,

  $\left|QP\right|=\left(a-b\right)\sin \theta$

Then,

  $\begin{array}{l}\left|PM\right|=\left|QM\right|-\left|QP\right|\\ \left|PM\right|=a\sin \theta -\left(a-b\right)\sin \theta \\ \left|PM\right|=a\sin -a\sin \theta +b\sin \theta \\ y=b\sin \theta \end{array}$ …… (2)

From equation (1) and (2).

  $\begin{array}{l}\frac{x}{a}=\cos \theta \\ \frac{y}{b}=\sin \theta \end{array}$

Then,

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$