Question

At what point on the paraboloid y = x² + z² is the tangent plane parallel to the plane

3x + 2y + 5z = 8?
(If an answer does not exist, enter DNE.)

Answer 1

To find the point on the paraboloid y = x² + z² where the tangent plane is parallel to the plane 3x + 2y + 5z = 8, we need to find the gradient vector of both surfaces and set them equal to each other. The point of tangency is (3/2, 1/2, 5/2).

Step-by-step explanation:

To find the point on the paraboloid y = x² + z² where the tangent plane is parallel to the plane 3x + 2y + 5z = 8, we need to find the gradient vector of both surfaces and set them equal to each other. The gradient vector of the paraboloid is ∇f = (2x, 1, 2z). The gradient vector of the plane is ∇g = (3, 2, 5).

Setting ∇f equal to ∇g, we get the equations:

2x = 3

1 = 2

2z = 5

Solving these equations, we find that x = 3/2, y = 1/2, and z = 5/2. Therefore, the tangent plane to the paraboloid at the point (3/2, 1/2, 5/2) is parallel to the plane 3x + 2y + 5z = 8.