We're given the paraboloid $z = x^2 + y^2$ and the ellipsoid $4x^2 + y^2 +z^2 = 9$, and we need to parameterize the curve at which they intersect.
I've tried parameterizing this trigonometrically, but it just never works. Substituting one equation into another seemed circular.
Substituting directly gives $3x^2 + z + z^2 = 9$, and completing the square, $3x^2 + (z + 1/2)^2 = 37/4$, so $x = \frac{\sqrt{37/4} \cos t}{\sqrt3}; z = \sqrt{37/4} \sin t - 1/2$.
Can you continue from here?