Convert the rectangular equation to polar form.Assume r 1. $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 9 \left( x ^ { 2 } - y ^ { 2 } \right)$
A) $r ^ { 2 } = 9 \cos 2 \theta$
B) $r ^ { 2 } = 9 \csc 2 \theta$
C) $r ^ { 2 } = 9 \cos \theta$
D) $r ^ { 2 } = 9 \sin 2 \theta$
E) $r ^ { 2 } = 9 \sin \theta$

Question

Convert the rectangular equation to polar form.Assume r > 1. $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 9 \left( x ^ { 2 } - y ^ { 2 } \right)$ 
A) $r ^ { 2 } = 9 \cos 2 \theta$
B) $r ^ { 2 } = 9 \csc 2 \theta$
C) $r ^ { 2 } = 9 \cos \theta$
D) $r ^ { 2 } = 9 \sin 2 \theta$
E) $r ^ { 2 } = 9 \sin \theta$

Quizplus · Accepted Answer

The given equation in rectangular coordinates is $(x^2 + y^2)^2 = 9(x^2 - y^2)$. In polar coordinates, $x^2 + y^2 = r^2$ and $x^2 - y^2 = r^2(\cos^2\theta - \sin^2\theta) = r^2\cos2\theta$. Substituting these into the given equation yields $r^4 = 9r^2\cos2\theta$, which simplifies to $r^2 = 9\cos2\theta$.

Convert the Rectangular Equation to Polar Form (x2+y2)2=9(x2−y2)\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 9 \left( x ^ { 2 } - y ^ { 2 } \right)(x2+y2)2=9(x2−y2)

Convert the Rectangular Equation to Polar Form $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 9 \left( x ^ { 2 } - y ^ { 2 } \right)$