From Cartesian to polar for a circle

Statement

Show that the Cartesian equation $x^2+y^2-4y = 0$ becomes $\rho = 4\sin\theta$ in polar coordinates. Interpret it geometrically.

Solution

Substitute $x^2+y^2 = \rho^2$ and $y = \rho\sin\theta$: $\rho^2 - 4\rho\sin\theta = 0 \;\Rightarrow\; \rho(\rho - 4\sin\theta) = 0.$ Excluding the pole $\rho=0$, we are left with $\boxed{\rho = 4\sin\theta}$. Geometrically, completing the square in the Cartesian equation: $x^2 + (y-2)^2 = 4$. It is the circle of centre $(0,2)$ and radius $2$, passing through the origin.

Links

Topics: Goniometry
Concepts: Polar coordinates · Polar equation · Cartesian plane
Methods: Polar equation
Skills: Interpreting a graph · Using formulae
Exercise types: Proof