Polar equation of the parabola

Statement

Find the polar equation of the parabola with Cartesian equation $y = x^2$.

Solution

Substitute $x = \rho\cos\theta$ and $y = \rho\sin\theta$ into the equation $y = x^2$: $\rho\sin\theta = (\rho\cos\theta)^2 = \rho^2\cos^2\theta.$ Excluding the pole ($\rho=0$, which corresponds to the origin, a point of the parabola) we divide by $\rho$: $\sin\theta = \rho\cos^2\theta \;\Rightarrow\; \boxed{\rho = \frac{\sin\theta}{\cos^2\theta}}.$ Equivalently, $\rho = \tan\theta\,\sec\theta$, valid for $\cos\theta\ne 0$.

Links

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