Question

Find equations of the following. 2(x − 3)2 + (y − 5)2 + (z − 4)2 = 10, (4, 7, 6)

(a) the tangent plane

Answer 1

The equation 2(x − 3)² + (y − 5)² + (z − 4)² = 10 represents a sphere. The equation of the tangent plane can be found by taking the gradient of the equation and substituting the coordinates of the given point.

Step-by-step explanation:

The equation 2(x − 3)² + (y − 5)² + (z − 4)² = 10 represents a sphere centered at the point (3, 5, 4) with a radius of √10. To find the equation of the tangent plane to this sphere at the point (4, 7, 6), we can use the formula for a plane: ax + by + cz = d.

First, we can find the normal vector to the sphere at the given point by taking the gradient of the equation. The gradient is given by (∂F/∂x, ∂F/∂y, ∂F/∂z) where F is the equation of the sphere.

Next, substitute the coordinates of the given point (4, 7, 6) and the normal vector into the equation of a plane to find the equation of the tangent plane.