Finding Parametric Equation of Tangent Line to Intersection of Surfaces?

[bodensee9]

Homework Statement

Hi, I need help with the following. I'm asked to find the parametric equation of the tangent line to the curve of the interrsection of the paraboloid z = x^2 + y^2 and the ellipsoid 4x^2+y^2+z^2 = 9 at the point (-1,1,2).

Homework Equations

I think I'm asked to find the gradient of the curves of the intersection, and then I know the vector with the direction of the line eof intersection, and then I can plug it back into find the parametric equation of the line.

The Attempt at a Solution

I guess I am trying to find the equation of the curve of intersection first? So, would I set the two equations equal so that I would get 4x^2+y^2+(x^2+y^2)^2 = 9? But then z has gone away? I guess I'm pretty clueless as to how to attempt to solve this problem?
Or if I'm right that the intersection is an ellipse, but how am I supposed to find the equation of the ellipse? Can anyone give me any pointers or hints?

 

[AlephZero]

You are only asked to find the tangent line at one particular point, so you don't need to find the equation of the complete intersection curve and then find its tangent. Doing all that would give you the answer, though.

Think about the tangent planes to the two surfaces at the point (-1,1,2).

 

[bodensee9]

Was going to post. Figured it out. You only need to take the cross product of the gradient to the two curves.