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

Question

Quizplus · Accepted Answer

To convert the given polar equation $r = 2(\sin\theta - \cos\theta)$ to rectangular coordinates, we use the relationships $x = r\cos\theta$ and $y = r\sin\theta$. Substituting $r = \sqrt{x^2 + y^2}$, we get $\sqrt{x^2 + y^2} = 2(y - x)$. Squaring both sides gives $x^2 + y^2 = 4y^2 - 8xy + 4x^2$, which simplifies to $x^2 + y^2 = 2y - 2x$, matching option D.

Find an Equivalent Equation in Rectangular Coordinates r=2(sin⁡θ−cos⁡θ)\mathrm { r } = 2 ( \sin \theta - \cos \theta )r=2(sinθ−cosθ)

Find an Equivalent Equation in Rectangular Coordinates $\mathrm { r } = 2 ( \sin \theta - \cos \theta )$