Chapter 11.6, Problem 6E

To determine

To find: the polar equation of conic.

Answer to Problem 6E

The polar equation of conic is $r=\frac{4}{2-\sin \theta }$ .

Explanation of Solution

Given information:

Ellipse, Focus: $\left(0,0\right)$ , eccentricity $\frac{1}{2}$ , directrix $y=-4$ .

Calculation: since, the directrix of the ellipse is $y=-4$ , so the standard polar equation of ellipse is,

  $r=\frac{ed}{1-e\sin \theta }$

Now, substitute in the standard polar equation and then simplify,

  $\begin{array}{l}r=\frac{ed}{1-e\sin \theta }\\ r=\frac{\frac{1}{2}\left(4\right)}{1-\left(\frac{1}{2}\right)\sin \theta }\text{ [substitute]}\\ r=\frac{2}{1-\frac{1}{2}\sin \theta }\text{ [divide]}\\ r=\frac{4}{2-\sin \theta }\text{ [simplify]}\end{array}$

Thus, the polar equation of ellipse is $r=\frac{4}{2-\sin \theta }$ .