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 given 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 $r = 18\cos\theta$ into $r\cos\theta = x$, we get $r = x$. Squaring both sides gives $r^2 = x^2$, and since $r^2 = x^2 + y^2$, we can substitute $18$ for $r$ to get $18^2 = x^2 + y^2$, which simplifies to $x^2 + y^2 = 324$. Rearranging $x^2 = 324 - y^2$ and taking the square root of both sides, we find that $x = 18$ or $x = -18$, but to match the options given, we recognize that $x = 18\cos\theta$ implies a circle centered on the x-axis. Thus, the correct transformation is $(x - 9)^2 + y^2 = 81$, which represents a circle with a radius of 9 units centered at $(9, 0)$, matching option A.

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 $r = 18 \cos \theta$