Find equations of the tangent line at the point (-2, 1, 5) to the hyperbola that is the intersection of the surface z = 2x² - 3y² and the plane z = 5. A) 3x + 4y = 2, z = 5 B) 4x + 3y = 5, z = -5 C) 4x + 3y = -5, z = 5 D) 4x - 3y = -5, z = -5 E) 3x + 4y = -2, z = 5

Question

Quizplus · Accepted Answer

The surface equation is z = 2x^2 - 3y^2. At the point (-2, 1, 5), the plane z = 5 intersects the surface, giving the equation 5 = 2(-2)^2 - 3(1)^2, which holds true. The gradient of the surface at this point is ∇f = (∂f/∂x, ∂f/∂y) = (8x, -6y). At (-2, 1), this is (-16, -6). The tangent line in the xy-plane is perpendicular to this gradient, giving the equation 4x + 3y = -5. The z-coordinate remains constant at z = 5.

Find Equations of the Tangent Line at the Point (-2