• Feb 8th 2007, 07:05 AM
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.
• Feb 8th 2007, 11:07 AM
Jenny20
Please check my work. The answer of this question is 2 lpl.
• Feb 8th 2007, 11:07 AM
CaptainBlack
Quote:

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