Replace the Cartesian equation with an equivalent polar equation.
-$$( x - 18 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 324$$
A) $r = 36 \cos \theta - 6 \sin \theta - 9$
B) $r ^ { 2 } = - 18 r \cos \theta + 3 r \sin \theta + 324$
C) $r ^ { 2 } = 36 r \cos \theta - 6 r \sin \theta - 9$
D) $\mathrm { r } ^ { 2 } = 36 \mathrm { r } \sin \theta - 6 \mathrm { r } \cos \theta - 9$

Question

Quizplus · Accepted Answer

In polar coordinates, $x=r\cos	heta$ and $y=r\sin	heta$. Substituting these, we get: $$(r\cos	heta - 18)^2+(r\sin	heta + 3)^2=324$$
Simplifying: $$r^2\cos^2	heta-36r\cos	heta+324+r^2\sin^2	heta+6r\sin	heta+9=324$$
Since $\cos^2	heta+\sin^2	heta=1$, we have: $$r^2-36r\cos	heta+6r\sin	heta-9=0$$
Factoring $r$ from the last two terms: $$r^2 = 36r\cos	heta - 6r\sin	heta + 9$$
Hence, the answer is (C).

Replace the Cartesian Equation with an Equivalent Polar Equation (x−18)2+(y+3)2=324( x - 18 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 324(x−18)2+(y+3)2=324

Replace the Cartesian Equation with an Equivalent Polar Equation $( x - 18 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 324$