Convert each equation to its rectangular form.
$r=\frac{2}{1-\sin\theta}$

• A. $y^2=4-4x$
• B. $x^2+y^2=2y$
• C. $y=\frac14x^2-1$
• D. $x^2-1=y$

Answer: (C)

Verified step by step guidance

1. Step 1: Start with the polar equation \( r = \frac{2}{1 - \sin\theta} \). Recall that the conversion from polar to rectangular coordinates involves using the relationships \( x = r\cos\theta \) and \( y = r\sin\theta \).
2. Step 2: Express \( \sin\theta \) in terms of \( y \) and \( r \) using the identity \( \sin\theta = \frac{y}{r} \). Substitute this into the equation to get \( r = \frac{2}{1 - \frac{y}{r}} \).
3. Step 3: Simplify the equation by multiplying both sides by \( r \) to eliminate the fraction: \( r^2 = 2r - 2y \).
4. Step 4: Substitute \( r^2 = x^2 + y^2 \) into the equation to convert it to rectangular form: \( x^2 + y^2 = 2r - 2y \).
5. Step 5: Rearrange the equation to isolate \( y \) and express it in terms of \( x \): \( y = \frac{1}{4}x^2 - 1 \).