Which answer is a rectangular form of the given polar equation?
$r = 18 \cos \theta$
A) $( x - 9 ) ^ { 2 } + y ^ { 2 } = 81$
B) $( x + 9 ) ^ { 2 } + y ^ { 2 } = 81$
C) $( x + 9 ) ^ { 2 } + y ^ { 2 } = 9$
D) $x ^ { 2 } + ( y + 9 ) ^ { 2 } = 81$
E) $x ^ { 2 } + ( y - 9 ) ^ { 2 } = 81$

Question

Quizplus · Accepted Answer

The polar equation $r = 18 \cos \theta$ can be converted to rectangular form using the relationship $r^2 = x^2 + y^2$ and $r\cos\theta = x$. Substituting $18\cos\theta$ for $r$ gives $x = 18\cos\theta$, so $r = x$. Squaring both sides gives $r^2 = x^2$, and substituting $r^2 = x^2 + y^2$ gives $x^2 + y^2 = 18^2$. Simplifying, we get $x^2 + y^2 = 324$, which can be rewritten as $(x - 0)^2 + y^2 = 18^2$. This doesn't directly match any of the options, but recognizing that $18^2 = 324$ and considering the standard form of a circle equation $(x - h)^2 + (y - k)^2 = r^2$, where $h$ and $k$ are the coordinates of the center and $r$ is the radius, we see that option A, $(x - 9)^2 + y^2 = 81$, is equivalent to $x^2 - 18x + 81 + y^2 = 81$, which simplifies to $x^2 + y^2 - 18x = 0$, or $x = 18\cos\theta$, matching the original polar equation when considering the relationship between $r$, $x$, and $\theta$. This explanation corrects the initial misinterpretation and identifies the correct conversion process from polar to rectangular coordinates, leading to the realization that none of the provided options directly match the correct conversion, indicating a mistake in the initial analysis.

Which Answer Is a Rectangular Form of the Given Polar r=18cos⁡θr = 18 \cos \thetar=18cosθ

Which Answer Is a Rectangular Form of the Given Polar $r = 18 \cos \theta$