Convert the rectangular equation (x² + y²)^(3/2) = 2xy to its polar form.
Correct: A
We use the standard conversions between rectangular and polar coordinates:
x = rcos(θ)
y = rsin(θ)
x² + y² = r²
Substitute these into the given rectangular equation:
(x² + y²)^(3/2) = 2xy
(r²)^(3/2) = 2(rcos(θ))(rsin(θ))
r^3 = 2r²cos(θ)sin(θ)
We know that 2sin(θ)cos(θ) = sin(2θ). So:
r^3 = r²sin(2θ)
To simplify, divide both sides by r². We must consider the case where r=0 separately. If r=0, then 0=0, so the pole is included. Dividing by r² assumes r ≠ 0, but the resulting equation usually covers the pole anyway.
r = sin(2θ)
The final answer is A.