The rectangular equation that corresponds to the polar equation $r ^ { 2 } = \cos \theta$ is

A) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$
B) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$
C) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$
D) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$
E) $\left( - x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$

Question

The rectangular equation that corresponds to the polar equation $r ^ { 2 } = \cos \theta$ is

A) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$
B) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$
C) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$
D) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$
E) $\left( - x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$

Quizplus · Accepted Answer

The polar equation $r^2 = \cos\theta$ can be converted to rectangular coordinates using the identities $r^2 = x^2 + y^2$ and $\cos\theta = \frac{x}{r}$. Substituting these into the given equation yields $x^2 + y^2 = x$, which, when rearranged, does not directly match any of the options. However, to match the form of the options provided, we take the given polar equation, multiply both sides by $r^3$ to get $r^5 = r^3\cos\theta$, which in rectangular coordinates translates to $(x^2 + y^2)^{\frac{5}{2}} = x(x^2 + y^2)^{\frac{3}{2}}$. This manipulation was to illustrate the process, but it seems there was a misunderstanding in the conversion process. Correctly, $r^2 = x^2 + y^2$ and $\cos\theta = x/r$, so substituting $r^2$ for $x^2 + y^2$ and multiplying both sides by $r$ to get $r^3 = x$, which translates to $(x^2 + y^2)^{\frac{3}{2}} = x$, aligns with option A. My explanation initially contained a calculation error; the correct conversion simply involves recognizing that $r^2 = x^2 + y^2$ and $r^2\cos\theta = x$, leading directly to the correct form without the additional manipulation I described.

The Rectangular Equation That Corresponds to the Polar Equation r2=cos⁡θr ^ { 2 } = \cos \thetar2=cosθ

The Rectangular Equation That Corresponds to the Polar Equation $r ^ { 2 } = \cos \theta$