Parametric equation of ellipse

Question

Find the curvature and the radius of curvature for (f) $x = 2 \cos t$ and $y = 3 \sin t$, $0 < t < 2\pi$ at point $(2, 0)$ and $(0, 3)$, where the parametric equation given is a ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ . May i know how to use the point given? I forgot the formula . thanks

Answer 1

$$\rho(x,y)=\left|\frac{(1+y'^2)^{3/2}}{y''(x)}|_{(x,y)}\right|$$

Here $$y=3\sqrt{1-x^2/4} \implies y'(x,y)=\frac{3x}{2\sqrt{1-x^2/4}}'$$ $$y''(x,y)=\frac{-6}{(4-x^2)^{3/2}}$$ so $$\rho(x,y)=\frac{(16+5x^2)^{3/2}}{48}$$ So $$\rho(2,0)=\frac{(16+20)^{3/2}}{48}=\frac{9}{2},$$ $$\rho(0,3)=\frac{4}{3}$$