Chapter 11.6, Problem 9E

To determine

To find: the polar equation of conic.

Answer to Problem 9E

The polar equation of conic is $r=\frac{10}{1+\sin \theta }$ .

Explanation of Solution

Given information:

parabola, Focus: $\left(0,0\right)$ , vertex at $\left(5,\frac{\pi }{2}\right)$ .

Calculation: since, the vertex of the parabola is $\left(5,\frac{\pi }{2}\right)$ , so the directrix of the parabola is $y=10$ ,

since, the directrix of the parabola is $y=10$ , so the standard polar equation is,

  $r=\frac{ed}{1+e\sin \theta }$

Since, the conic is parabola so $e=1$ ,

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

  $\begin{array}{l}r=\frac{ed}{1+e\sin \theta }\\ r=\frac{\left(1\right)10}{1+\left(1\right)\sin \theta }\text{ [substitute]}\\ r=\frac{10}{1+\sin \theta }\text{ [simplify]}\end{array}$

Thus, the polar equation of parabola is $r=\frac{10}{1+\sin \theta }$ .