Question

Find an equation of the surface satisfying the conditions. The set of all points equidistant from the point (0, 2, 0) and the plane y = −2.

A) Equation: Fill in the equation.
B) Surface: Identify the surface type.

Answer 1

The equation of the surface satisfying the given conditions is (y + 2)^2 = 16x. The surface type is a parabolic cylinder.

Step-by-step explanation:

To find an equation of the surface satisfying the conditions, we need to find all points that are equidistant from the point (0, 2, 0) and the plane y = -2. The set of all points equidistant from a point and a plane form a parabolic cylinder. The equation of this surface can be written as (y - y0)^2 = 4a(x - x0), where (x0, y0, z0) is the point on the plane closest to the given point and a is the distance between the point and the plane.

In this case, the point on the plane closest to (0, 2, 0) is (0, -2, 0), and the distance between (0, 2, 0) and the plane is 4 units. Therefore, the equation of the surface is (y + 2)^2 = 16x.