# Thread: radius of curvature

1. ## radius of curvature

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.

2. 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.

3. 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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# radius of curvature of parabola

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