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I need to find the parametric equations for the the tan line at point, P(x1,y1,z1) on the curve formed from paraboloid intersection with ellipsoid.

The parametric equations for the level surfaces that make up paraboloid and ellipsoid are NOT given.

The level functions for paraboloid and the level function for ellipsoid are given.

This is what I've done so far.

I found the equation of the curve that forms from the intersection.

[c(x,y)] = curve of paraboloid and ellipsoid intersection.

The tangent vector at p(x1,y1,z1) on curve should be the same as the tangent vector at same point on paraboloid and ellipsoid.

I have taken the gradient of curve dot tangent vector = 0

I have taken the gradient of level fxn of paraboloid dot tangent vector = 0

I have taken the gradient of level fxn of ellipsoid dot tangent vector = 0

Solving the three linear equations that all equal to zero gets me nowhere.

I am stuck. Help me.