Parametrize the curve of intersection of the sphere x^2+y^2+z^2=1 and the elliptic cylinder x^2+2z^2=1. Find the total length of this intersection curve.

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- Dec 7th 2009, 02:37 PMjonnyp2009Parametrize the curve of intersection
Parametrize the curve of intersection of the sphere x^2+y^2+z^2=1 and the elliptic cylinder x^2+2z^2=1. Find the total length of this intersection curve.

- Dec 7th 2009, 02:54 PMTheEmptySet

Note you can solve the 2nd equation for $\displaystyle z^2$ to get

$\displaystyle z^2=\frac{1-x^2}{2}$

sub this into the first equation to get

$\displaystyle x^2+y^2+\frac{1}{2}-\frac{x^2}{2}=1$

$\displaystyle \frac{x^2}{2}+y^2=\frac{1}{2} \iff x^2+\frac{y^2}{(\frac{1}{\sqrt{2}})^2}=1$

This is an equation of an ellipse...

This should get you started - Dec 8th 2009, 07:17 AMjonnyp2009
ok thanks i got it now, i then get x=cost and solve for y then plug them both into sphere equation for z

so $\displaystyle y=1/sqrt(2)sint$

and $\displaystyle z=1/sqrt(2)sint$