Finding a geodesic on a plane using polar coordinates

Question

This is from my homework on PDE.

I need to find a geodesic on a plane using polar coordinates. Now, I know

$dl^2 = x^2+y^2$ hence $l=\int \sqrt{dx^2+dy^2}$,

but I get stuck while converting coordinates. I guess it's the result of a hole in education. Nevertheless, could you help me?

Answer 1

In polar coordinates the arc length is given by: $$l=\int_{a}^{b}\sqrt{[r(\theta)^2+(\frac{dr(\theta)}{d\theta})^2}d\theta$$