Convert the polar equation r = 4 / (2 - cos(θ)) to its rectangular form.
Correct: A
We use the standard conversions between polar and rectangular coordinates:
x = rcos(θ)
y = rsin(θ)
r = √(x² + y²)
Given the polar equation:
r = 4 / (2 - cos(θ))
Multiply both sides by (2 - cos(θ)):
2r - rcos(θ) = 4
Substitute r = √(x² + y²) and rcos(θ) = x:
2√(x² + y²) - x = 4
Isolate the square root term:
2√(x² + y²) = 4 + x
Square both sides to eliminate the square root:
(2√(x² + y²))² = (4 + x)²
4(x² + y²) = 16 + 8x + x²
Distribute and rearrange terms to form a general conic equation:
4x² + 4y² = 16 + 8x + x²
4x² - x² - 8x + 4y² - 16 = 0
3x² - 8x + 4y² - 16 = 0
This is the equation of an ellipse.
The final answer is A.