# Parametric equation of ellipse

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

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– user284001
Jan 20 at 11:49
• sorry about that Jan 20 at 11:51

$$\rho(x,y)=\left|\frac{(1+y'^2)^{3/2}}{y''(x)}|_{(x,y)}\right|$$
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 $$\rho(2,0)=\frac{(16+20)^{3/2}}{48}=\frac{9}{2},$$ $$\rho(0,3)=\frac{4}{3}$$