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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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    $\begingroup$ Can you add your own efforts here? Let us know what you don't understand, as this is not a homework site, without these efforts the number of responses will be very low. $\endgroup$
    – user284001
    Jan 20 at 11:49
  • $\begingroup$ sorry about that $\endgroup$
    – Blinkeu5
    Jan 20 at 11:51
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$$\rho(x,y)=\left|\frac{(1+y'^2)^{3/2}}{y''(x)}|_{(x,y)}\right|$$

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 $$\rho(2,0)=\frac{(16+20)^{3/2}}{48}=\frac{9}{2},$$ $$\rho(0,3)=\frac{4}{3}$$

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  • $\begingroup$ Thank you so much. I appreciate it :). Thanks again !! $\endgroup$
    – Blinkeu5
    Jan 20 at 14:14

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