It can be shown that the polar curve r=asin(θ)+bcos(θ), where ab≠0, is a circle. Find the center (in Cartesian coordinates) and radius

Question

It can be shown that the polar curve r=asin(θ)+bcos(θ), where ab≠0, is a circle. Find the center (in Cartesian coordinates) and radius of this circle in terms of a and b.

x-coordinate of center:

y-coordinate of center:

Radius:

Answer 1

r = asinθ + bcosθ

 

So, r² = arsinθ + brcosθ

 

x²+y² = ay + bx

 

(x² - bx) + (y² - ay) = 0

 

(x² - bx + b²/4) + (y² - ay + a²/4) = (a²+b²)/4

 

(x-b/2)² + (y-a/2)² = [√(a²+b²)/2]²

 

    Center = (b/2, a/2)    Radius = √(a²+b²)/2