how would you convert this to a polar equation: (x-1)^2 + (y+2)^2 =5 ?

Question

I know the radius is sqrt 5, and is center (1,2)?

Answer 1

(x-1)^2+(y+2)^2=5

x^2-2x+1+y^2+4y+4=5

x^2+y^2=r^2

r^2-2x+4y+5=5

r^2-2x+4y=0

x=rcosθ and y=rsinθ

r^2-2rcosθ+4rsinθ=0

r^2=2rcosθ-4rsinθ

r=2cosθ-4sinθ

Answer 2

Use the relationships:

x = r*cosθ

y = r*sinθ

Substitute those in for x and y to convert to polar form. Polar form will look like r = (function of θ), no x's or y's - only r and θ.

(x - 1)² + (y + 2)² = 5 becomes

(r*cosθ - 1)² + (r*sinθ + 2)² = 5

Multiply out and simplify to get:

r²*cos²θ - 2r*cosθ + r²*sin²θ + 4r*sinθ = 0

Group the two r² terms together and factor out r² to get:

r²*(cos²θ + sin²θ) - 2r*cosθ + 4r*sinθ = 0

The term cos²θ + sin²θ = 1 by the Pythagorean Identity so the expression becomes:

r² - 2r*cosθ + 4r*sinθ = 0

Factor out r to get:

r(r - 2cosθ + 4sinθ) = 0

Set each piece equal to zero to get:

r = 0 or

r - 2cosθ + 4sinθ = 0

r = 0 is not a solution that is meaningful so use r - 2cosθ + 4sinθ = 0

Solve for r to get:

r = 2cosθ - 4sinθ

Answer 3

You are correct about the radius, but the general form for the equation of a circle is (x-h)^2+(x-k)^2=r^2 . So the center is (1,-2).

Answer 4

the center is (1, 2), the radius is sqrt 5. Then the circle equation is (x-1)^2+(y-2)^2=5.