Chapter 9, Problem 7CT

To determine

The rectangular equation of polar equation $r{\sin }^2\theta +8\sin \theta =r$ and graph the equation.

Answer to Problem 7CT

Solution:

The rectangular equation of polar equation $r{\sin }^2\theta +8\sin \theta =r$ is $8y=x^2$.

The graph of the equation $r{\sin }^2\theta +8\sin \theta =r$ is,

Explanation of Solution

Given information:

The polar equation, $r{\sin }^2\theta +8\sin \theta =r$

Explanation:

Consider the equation $r{\sin }^2\theta +8\sin \theta =r$.

Multiply both sides by $r$,

$r^2{\sin }^2\theta +8r\sin \theta =r^2$

In polar coordinates $x=r\cos \theta ,y=r\sin \theta$ and $r^2=x^2+y^2$.

Therefore, the equation becomes

$y^2+8y=x^2+y^2$.

$\Rightarrow 8y=x^2$

Therefore, the equation in rectangular coordinates is $8y=x^2$.

The standard equation of parabola is $4p\left(y-k\right)={\left(x-h\right)}^2$ with vertex at $\left(h,k\right)$ and focal length $\left|p\right|$.

Compare the equation $4\left(2\right)\left(y-0\right)={\left(x-0\right)}^2$ with standard equation of circle.

Thus, the equation $8y=x^2$ is an equation of parabola with center $\left(0,0\right)$ and focal length $2$.

The graph of the equation $r{\sin }^2\theta +8\sin \theta =r$ is,