Find a polar equation for the circle.
-$$x ^ { 2 } + \frac { 8 } { 3 } x + y ^ { 2 } = 0$$
A) $r = \frac { 8 } { 3 } \sin \theta$
B) $r = - \frac { 8 } { 3 } \sin \theta$
C) $r = \frac { 8 } { 3 } \cos \theta$
D) $r = - \frac { 8 } { 3 } \cos \theta$

Question

Quizplus · Accepted Answer

To get a polar equation, we substitute $x=r\cos	heta$ and $y=r\sin	heta$:
$$r^2\cos^2	heta+\frac{8}{3}r\cos	heta+r^2\sin^2	heta=0$$
Simplify and combine like terms:
$$r^2+\frac{8}{3}r\cos	heta=0$$
$$r(r+\frac{8}{3}\cos	heta)=0$$
Therefore, $r=0$ or $r=-\frac{8}{3}\cos	heta$. This is the equation of a circle with center at the origin and radius $\frac{8}{3}$, so we need to use the absolute value of $r$ to get the correct answer. Hence, the polar equation is $r=\frac{8}{3}|\cos	heta|=-\frac{8}{3}\cos	heta$ when $\cos	heta<0$, which corresponds to answer choice D.

Find a Polar Equation for the Circle x2+83x+y2=0x ^ { 2 } + \frac { 8 } { 3 } x + y ^ { 2 } = 0x2+38​x+y2=0

Find a Polar Equation for the Circle $x ^ { 2 } + \frac { 8 } { 3 } x + y ^ { 2 } = 0$