Answer
$\kappa(t)= \dfrac{ab}{(a^2 \sin^2 t+b^2 \cos^2 t)^{3/2}}$
Work Step by Step
The curvature $\kappa$ for a plane curve system is:
$\kappa (t)= \dfrac{||r'(t) \times r''(t)||}{||r'(t)||^3}$
We have:
$r'(t) =\lt -a \sin t, b \cos t \gt$ and $r''(t) =\lt -a \cos t, b \sin t \gt$
Thus,
$\kappa(t) = \dfrac{||-ab k ||}{|| \lt -a \cos t, b \sin t \gt ||^3} \\=\dfrac{ab}{[a^2 \sin^2 t+b^2 \cos^2 t]^{3/2}}$
Therefore,
$\kappa(t)= \dfrac{ab}{(a^2 \sin^2 t+b^2 \cos^2 t)^{3/2}}$
