Find the slope of the tangent line to the curve of intersection of the
elliptic paraboloid z = x^2 + (y^2)/4 and the plane x = 2 at the point
(2,2,5).

Google Answers discourages and may remove questions that are homework
or exam assignments.

Let's look at the curve of intersection first. Obviously we'll have
x=2, and z = 4 + y^2 / 4. I assume the 'slope' refers to the slope in
the y-z plane, since it's pretty meaningless otherwise.

Now, z'(y) (or dz/dy if you prefer that notation) = 2y / 4 = y / 2
along this curve. So at the point (2, 2, 5) (which is indeed on the
curve - you need to check these things) z'(y) = 1. Now the slope is an
angle t such that tan t = z'(y), so it's pi/4 or 45 degrees.