How do you convert (x-3)^2+y^2=9 into polar coordinates?

Answer 1

To convert Cartesian coordinates (x and y), or in this case, an equation with x and y, you need to know that x = r * cos θ and y = r * sin θ.

Think of a point on a graph. You are standing at the origin and you can either give directions to the point in terms of x and y or a straight line distance r where θ is the angle between the path that you're walking and the horizontal axis (which marks 0º on the polar plane).

So plug in the formulas above and you get

(x-3)²+y²=9

(r * cos θ-3)² + (r * sin θ)² = 9

r² cos² θ - 6 cos θ + 9 + r² sin² θ = 9

you need to know that sin² θ + cos² θ = 1

r²(sin² θ + cos² θ) - 6 cos θ = 0

r² - 6 cos θ = 0

r = 6 cos θ

Now, if you were to graph this in a polar plane, you would get the same circle you'd get graphing (x-3)²+y²=9 on a Cartesian plane.

Answer 2

Multiplying out the expression gives x² -6x +9 +y² = 9 substituting in the polar equivalents gives (a = theta)

r²cos²(x) -6rcos(a) +9 + r²sin²(a) = 9 So, 1 -6rcos(a) = 0 or r = 1/(6cos(a))