radius of curvature

**Jenny20** · February 8th 2007, 07:05 AM

Find the radius of curvature of the parabola y^2 = 4px at (0,0).

Please teach me how to solve this question. Thank you very much.

**Jenny20** · February 8th 2007, 11:07 AM

Please check my work. The answer of this question is 2 lpl.
I got the answer is 2p. Could you please help me find out the reason? Thank you very much.

**CaptainBlack** · February 8th 2007, 11:07 AM

Originally Posted by Jenny20

Find the radius of curvature of the parabola y^2 = 4px at (0,0).

Please teach me how to solve this question. Thank you very much.

In this case take the curvature as:

$ \kappa=x''/(1+x')^{3/2}\, $

where we consider $x\,$ as a function of $y\,$ and differentiation is with respect to $y\,$

This has $x\,$ and $y\,$ interchanged compared to the usual expression but the absolute value of the curvature is the same either way, and this way is easier to handle.

Here:

$x=y^2/(4p)\,$

so:

$\frac{dx}{dy}=x'=y/(2p)\,$ ,

and:

$ \frac{d^2x}{dy^2}=x''=1/(2p)\, $

so at $(0, 0)\,$ , $x'=0\,$ , so:

$ \kappa=\frac{[1/(2p)]}{1}=1/(2p)\, $

so the radius of curvature $r=|1/\kappa|=2|p|\,$ .

RonL

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