Consider x = h(y,z) as a parametrized surface, equation of tangent plane to the surface at the point...?

Question

Consider x = h(y,z) as a parametrized surface Φ(y,z) in the natural way. Write the equation of the tangent plane to the surface at the point (2, 2, 0) [with the coefficient of x being 1] given that
∂h/∂y(2,0) = -5 and ∂h/∂z (2,0) = 0.


_____________ = 0

Any help would be great! Answer and explanation of the steps to the answer. Thanks

Answer 1

The general case has the surface as F(x,y,z) = 0. Then the gradient of F at a point is normal to the tangent plane at that point. For the special case in this problem, the form of F is

F = x - h(y,z) = 0 so

grad F = < 1, -∂h/∂y, -∂h/∂z >

At the point (2,2,0), which I will denote P, this is <1, 5, 0>

P of course lies in the tangent plane. Now let Q(x,y,z) be any other point in the plane. Then the vector PQ is perpendicular to grad F evaluated at P, that is, to <1,5,0>. So the dot product of PQ and <1,5,0> is 0:

• <1,5,0> = 0
1(x-2) + 5(y-2) + 0(z-2) = 0
x + 5y -12 = 0