Cartesian equations for the line tangent to two surface.

Question

I am asked to find a Cartesian equation for the line tangent to both the surfaces

x^2+y^2+2z^2=4 and z=e^(x-y) at the point (1.1.1)

I tried to find out normal vector to both surfaces and tangent plane at the point (1.1.1)

and it does not help me at all;;

Answer 1

You can find the tangent planes to both curves at (1,1,1), then take the cross product of the two normal vectors to get the direction vector of the tangent line, since this will give you a vector that lies on both tangent planes.