What you have done is perfectly valid.
Question: The intersection of the plane and the surface is an ellipse E. Find the points on E that are closest to the origin.
What I did:
First I join the two equations together to form:
Using and treating the above equation as a constraint:
I managed to get:
Subst back into the equation I have respectively.
So the points I conclude is
Is this right? Or do I have to form two contraint equations instead?