Question

Let the polar coordinates of the point (x, y) be (r, θ). Determine the polar coordinates for the point (2x, y)

Answer 1

For (x,y), we have
r^2 = x^2 + y^2
tanθ = y/x
Let the new point have polar coordinates (p,φ). Then
p^2 = (2x)^2 + y^2 = 4x^2 + y^2 = r^2 + 3x^2
tanφ = y/(2x) = 1/2 (y/x) = 1/2 tanθ

You can massage those into other forms, of course.