Question

Find a polar equation for the circle with center at the point $(0,4)$and radius 4. Graph this circle.

Short answer

The equation is $r=8\sin \theta$and the graph is

Step-by-step solution

Step 1. Given Information 

The given center is$(0,4)$and radius is 4.

Step 2. Explanation 

The equation has a circle in rectangular cartesian coordinates is given below,

${(x-h)}^2+{(y-k)}^2=a^2$

The circle has a radius 4 units and center at $(0,4)$,

$$\begin{gathered} {(x-0)}^2+{(y-4)}^2=4^2 \\ {x}^2+y^2-8y=0 \end{gathered}$$

Thus, equation of circle in rectangular coordinates is $x^2+y^2-8y=0$

Substitute the values $x=r\cos \theta ,y=r\sin \theta$to get the polar form of the equation.

$$\begin{gathered} {\left(r\cos \theta \right)}^2+{\left(r\sin \theta \right)}^2-8\left(r\sin \theta \right)=0 \\ {r}^2-8r\sin \theta =0 \\ r=8\sin \theta \end{gathered}$$

And the graph is as follows,