Find an equivalent equation in rectangular coordinates.
-$\mathrm { r } - 5 \sin \theta = 5 \cos \theta$
A) $\sqrt { x ^ { 2 } + y ^ { 2 } } - 5 y = 5 x$
B) $x ^ { 2 } + y ^ { 2 } = 0$
C) $x ^ { 2 } + y ^ { 2 } - 5 y = 5 x$
D) $x ^ { 2 } + y ^ { 2 } + 5 y = 5 x$

Question

Quizplus · Accepted Answer

To convert the given polar equation $r - 5\sin\theta = 5\cos\theta$ to rectangular coordinates, we use the relationships $r = \sqrt{x^2 + y^2}$, $\sin\theta = \frac{y}{r}$, and $\cos\theta = \frac{x}{r}$. Substituting these into the given equation yields $\sqrt{x^2 + y^2} - 5\frac{y}{\sqrt{x^2 + y^2}} = 5\frac{x}{\sqrt{x^2 + y^2}}$. Multiplying through by $\sqrt{x^2 + y^2}$ gives $x^2 + y^2 - 5y = 5x$, which matches choice C.

Find an Equivalent Equation in Rectangular Coordinates r−5sin⁡θ=5cos⁡θ\mathrm { r } - 5 \sin \theta = 5 \cos \thetar−5sinθ=5cosθ

Find an Equivalent Equation in Rectangular Coordinates $\mathrm { r } - 5 \sin \theta = 5 \cos \theta$