I have the ellipse defined by the equation $\frac{x^2}{2} + \frac{z^2}{8} = 1$, and I need to find the parametric equation for a line tangent to the ellipse at $(1,2,2)$
I know that the parametric equation for the ellipse is $x = 2cos(t), y=2, z=8sin(t)$, but I'm not sure what to do next.
The original question was
Find parametrization of curve of intersection of following two surfaces
Surface
$4x^2+2y^2+z^2=16$ intersects plane
$y = 2$
and generates an ellipse
Find the parametric equation of the tangent line to this ellipse at
( 1 , 2 , 2 )
Best Answer
Intersection of the given surface and the plane is,
$\displaystyle \frac{x^2}{2} + \frac{z^2}{8} = 1, y = 2$ which can be parametrized as
$r(t) = \langle\sqrt2 \ cos (t), 2, \sqrt8 \ sin (t) \ \rangle $
So $r'(t) = \langle - \sqrt2 \ sin (t), 0, \sqrt8 \ cos (t) \ \rangle$
At $(1, 2, 2), x = \sqrt2 \cos (t) = 1, z = \sqrt 8 \sin (t) = 2 \implies t = \frac{\pi}{4}$.
$r'(t = \frac{\pi}{4}) = (-1, 0, 2)$
So the equation of tangent line is $(1,2, 2) + (-1, 0, 2) t$
or $ \ x = 1 - t, y = 2, z = 2 + 2t$