A group of people, including myself, have been trying to figure out how to go about solving for d in polar coordinate form for a certain problem

Question

If there is a circle expressed on the Cartesian plane as (x-2)^2 + y^2 = 9, how do you go about expressing the same equation in polar coordinates in the form ar^2 + br(cos(theta)) + cr( sin(theta) ) + d = 0? We already know that d equals -5, but we have trouble figuring out everything else out.

Answer 1

(x-2)² + y² = 9

x² - 4x + 4 + y² = 9

Let x = rcosθ and y = rsinθ

Then, r²cos²θ + r²sin²θ - 4rcosθ - 5 = 0

r²[cos²θ + sin²θ] - 4rcosθ - 5 = 0

r² - 4rcosθ - 5 = 0

The equation above has the form ar² + br(cosθ) + cr(sinθ) + d = 0, where a = 1, b = -4, c = 0, and d = -5.