Convert the polar equation to rectangular form. $r ^ { 2 } = 2 \sin \theta$
A) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 2 } y ^ { 2 / 3 } = 0$
B) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
C) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 2 } y ^ { 2 / 3 } = 0$
D) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
E) $x ^ { 2 } - y ^ { 2 } - \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$

Question

Convert the polar equation to rectangular form. $r ^ { 2 } = 2 \sin \theta$ 
A) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 2 } y ^ { 2 / 3 } = 0$
B) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
C) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 2 } y ^ { 2 / 3 } = 0$
D) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
E) $x ^ { 2 } - y ^ { 2 } - \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$

Quizplus · Accepted Answer

The correct conversion of $r^2 = 2\sin\theta$ to rectangular form involves recognizing that $r^2 = x^2 + y^2$ and $\sin\theta = \frac{y}{r}$. Substituting these into the original equation gives $x^2 + y^2 = 2\frac{y}{\sqrt{x^2 + y^2}}$. However, none of the provided options directly match the correct conversion process due to a mistake in the interpretation of the options. The correct conversion should result in an equation involving $x^2 + y^2$ and a term with $y$, but none of the options accurately reflect the standard conversion process. The selection was based on a misunderstanding of the conversion process and the options provided.

Convert the Polar Equation to Rectangular Form r2=2sin⁡θr ^ { 2 } = 2 \sin \thetar2=2sinθ

Convert the Polar Equation to Rectangular Form $r ^ { 2 } = 2 \sin \theta$