how to calculate the curvature of an ellipse

We have $\alpha'(t) = \langle -a \sin t, b \cos t \rangle$ and $\alpha''(t) = \langle -a\cos t , - b \sin t\rangle$, thus $|\alpha'(t)| = \sqrt{a^2\sin^2t + b^2\cos^2t}$. We have that $T(t) = \frac{\alpha'(t)}{|\alpha'(t)|},$ which has length $1$ and is tangent to $\alpha (t).$ $$T(t) = \langle \frac{-a \sin t}{\sqrt{a^2\sin^2t + b^2\cos^2t}}, \frac{b\cos t}{\sqrt{a^2\sin^2t + b^2\cos^2t}}\rangle$$ which leads to $$\kappa = \frac{|T'(t)|}{|\alpha'(t)| }= \frac{ab}{(\sqrt{a^2\sin^2t + b^2\cos^2t})^3}.$$