finding a vector valued function for the intersection of two shapes

Question

I have a problem for my cal 3 class to find a vector valued function for the intersection of these two equations.

$4x^2+4y^2+z^2=16$
and
$x=z^2$

so i know that the first equation is a ellipsoid and the second is a folded sheet or cylinder. I also believe that I need to use the parametric equation for an ellipsoid to come up with my final vector valued equation. I know i can get theta by putting the $4y^2$ over $4x^2$ and taking the arc sign, but I'm having trouble perameterising these and then uniting them with a common t. An explanation for how to setup and solve this problem would be a great help.

Answer 1

Using the information provided,$5z^2 +4y^2 = 16$, which is an ellipse in the $y-z$ plane. You can now use parameterize this curve using trig functions.

Answer:

$$\dfrac{5z^2}{16} + \dfrac{y^2}{4} = 1 $$ $\implies$ $$ \mathbf{r}(t) = \left<y(t),z(t)\right> = \left< 2\cos(t), \dfrac{4}{\sqrt{5}} \sin(t) \right>$$