Ellipsoid has equation $\displaystyle x^2+y^2-yz+z^2=1$

It fits inside a cylinder, which if I did the math correctly, is $\displaystyle x^2+y^2-\frac{y^2}{2}+\frac{y^2}{4}=1$ Parameterize the curve of intersection of the ellipsoid with the elliptic cylinder.

So I'm looking for a function that describes where the ellipsoid touches the cylinder?

I'm not sure where to start