Tangent Line to Curve of Intersection: Calculating the Normal and Cross Product

[Yitin]

Homework Statement

Find parametric equations for the tangent line to the curve of intersection of the cone z=√(x2 + 4y2) and the plane 3z = x + 2y + 8 at the point (3,2,5)

2. The attempt at a solution

I was trying to make the two Zs equal to each other, and solve for x or y, but I couldn't get any of them separate. I tried squaring both of them so they would both have things like X². It didn't work out.

 

[LCKurtz]

Homework Statement

Find parametric equations for the tangent line to the curve of intersection of the cone z=√(x² + 4y²) and the plane 3z = x + 2y + 8 at the point (3,2,5)

2. The attempt at a solution

I was trying to make the two Zs equal to each other, and solve for x or y, but I couldn't get any of them separate. I tried squaring both of them so they would both have things like X². It didn't work out.

That's enough reason to try a different approach. Remember that the curve of intersection of two surfaces lies in both surfaces. So if you calculate the normal to each surface at a point on the surface, each normal will be perpendicular to the curve at that point. So the cross product of the normals will be tangent to the curve. And gradients are perpendicular to surfaces...