Question

Show that the equation of the tangent plane to the elliptic paraboloid z/c = x²/a² + y²/b² at the point (x₀, y₀, z₀) can be written as

Answer 1

The equation of the tangent plane to the elliptic paraboloid can be written as (2x₀x/a²) + (2y₀y/b²) + (z - z₀) = 0.

Step-by-step explanation:

To find the equation of the tangent plane to the elliptic paraboloid at a given point, we first need to find the partial derivatives of the equation with respect to x and y. Taking the partial derivatives of the equation z/c = x²/a² + y²/b², we get ∂z/∂x = (2x/a²) and ∂z/∂y = (2y/b²). Next, we use these derivatives to find the equation of the tangent plane using the point (x₀, y₀, z₀). The equation of the tangent plane is given by (x - x₀)(∂z/∂x) + (y - y₀)(∂z/∂y) + (z - z₀) = 0. Substituting the partial derivatives and the given point, the equation of the tangent plane can be written as (2x₀x/a²) + (2y₀y/b²) + (z - z₀) = 0.