Convert the polar equation to rectangular form. $r ^ { 2 } = 4 \sin \theta$
A) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 16 } y ^ { 2 / 3 } = 0$
B) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 16 } y ^ { 2 / 3 } = 0$
C) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
D) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
E) $x ^ { 2 } - y ^ { 2 } - \sqrt [ 3 ] { 16 } y ^ { 2 / 3 } = 0$

Question

Convert the polar equation to rectangular form. $r ^ { 2 } = 4 \sin \theta$ 
A) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 16 } y ^ { 2 / 3 } = 0$
B) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 16 } y ^ { 2 / 3 } = 0$
C) $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
D) $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 4 } y ^ { 2 / 3 } = 0$
E) $x ^ { 2 } - y ^ { 2 } - \sqrt [ 3 ] { 16 } y ^ { 2 / 3 } = 0$

Quizplus · Accepted Answer

The given polar equation is $r^2 = 4\sin\theta$. In rectangular coordinates, $r^2 = x^2 + y^2$ and $\sin\theta = \frac{y}{r}$. Substituting these into the given equation, we get $x^2 + y^2 = 4\frac{y}{\sqrt{x^2 + y^2}}$. This equation simplifies to $x^2 + y^2 = 4y$, or $x^2 + y^2 - 4y = 0$. The provided options do not directly match this simplified form, indicating a potential misunderstanding in the conversion process. The correct conversion from polar to rectangular form should yield an equation involving $x$ and $y$ that represents the same curve, but none of the provided options correctly represent the conversion of $r^2 = 4\sin\theta$ under standard transformations. Therefore, the explanation points out a mistake in the interpretation of the options relative to the expected conversion process.

Convert the Polar Equation to Rectangular Form r2=4sin⁡θr ^ { 2 } = 4 \sin \thetar2=4sinθ

Convert the Polar Equation to Rectangular Form $r ^ { 2 } = 4 \sin \theta$