Chapter 9.7, Problem 50E

To determine

To calculate:

The polar equation of the conic by using the following data:

Conic section: Ellipse

Vertices: $\left(20,0\right),\left(4,\pi \right)$

Answer to Problem 50E

The polar equation of the ellipse is derived as $r=\frac{20}{3-2\cos \theta }$ .

Explanation of Solution

Given information:

Conic section: Ellipse

Vertices: $\left(20,0\right),\left(4,\pi \right)$

Formula used:

The equation of the ellipse is given by the following expression:

  $r=\frac{ep}{1-e\cos \theta }$

Where,

Eccentricity: $e$

Distance from the focus to pole: $\left|p\right|$

The distance between the focus and vertex is given by the following expression:

  $a-ae$

The distance between the vertices is the length major axis $\left(2a\right)$

Calculation:

Here, the given conic section is ellipse and its vertices are $\left(20,0\right),\left(4,\pi \right)$ . Now need to find the equation of ellipse:

From the given data, get that horizontal axis is the major axis. So, the nearest vertex to pole is on the left side of the pole and vertical axis represents the directrix.

So the equation of the ellipse is given by the following expression:

  $r=\frac{ep}{1-e\cos \theta }$

The length major axis $\left(2a\right)$ is calculated as:

  $\begin{array}{l}2a=24\\ a=12\end{array}$

Since, the nearest vertex point lies at distance of $4units$ from the pole.

So, distance between the focus and vertex is calculated as:

  $\begin{array}{l}a-ae=4\\ 12-12e=4\\ e=\frac{2}{3}\end{array}$

Now, distance from the focus to directrix is calculated as:

  $\begin{array}{l}p=\frac{a}{e}-ae\\ p=\frac{12}{\left(\frac{2}{3}\right)}-12\left(\frac{2}{3}\right)\\ p=10\end{array}$

Hence, the equation of the ellipse is derived as:

  $r=\frac{ep}{1-e\cos \theta }$

Put the calculated values:

  $\begin{array}{l}r=\frac{ep}{1-e\cos \theta }\\ r=\frac{\left(\frac{2}{3}\right)\left(10\right)}{1-\left(\frac{2}{3}\right)\cos \theta }\\ r=\frac{20}{3-2\cos \theta }\end{array}$