Chapter 12, Problem 9CR

To determine

To find: The polar equation of a circle.

Answer to Problem 9CR

The polar equation of the circle passing through the pole and centered at $\left(0,4\right)$ is $r=8\sin \theta$ .

Explanation of Solution

Given information:

The given points is,

Center at $\left(0,4\right)$

Calculation:

The equation of a circle with radius $4$ and center at $\left(0,4\right)$ is $x^2+{\left(y-4\right)}^2=4^2$ .

Simplify the equation,

  $\begin{array}{c}{x}^2+{\left(y-4\right)}^2=4^2\\ x^2+y^2-8y+16=16\\ x^2+y^2-8y=0\end{array}$

Substitute the values of $r^2$ for $x^2+y^2$ and $r\sin \theta$ for $y$ .

  $\begin{array}{c}{r}^2-8\left(r\sin \theta \right)=0\\ r=8\sin \theta \end{array}$

The polar form of $x^2+{\left(y-4\right)}^2=4^2$ , is $r=8\sin \theta$ .

Therefore, the polar equation of the circle passing through the pole and centered at $\left(0,4\right)$ is $r=8\sin \theta$ .