Find an equation for the surface consisting of all points that are equidistant from the point and the plane .

Question

Find an equation for the surface consisting of all points that are equidistant from the point  and the plane  .

Quizplus · Accepted Answer

The surface equidistant from a point and a plane is a parabolic cylinder. The equation of the parabolic cylinder can be written as (x-a)^2 = 4p(y-b), where (a,b) is the vertex and p is the distance from the vertex to the focus. In this case, the vertex is the point equidistant from the given point and plane, which is the intersection of the line perpendicular to the plane at the given point and the line passing through the given point and the center of the plane. Solving for the vertex, we get (0,0,0). The distance from the vertex to the focus, which is the given point, is the same as the distance from the vertex to the directrix, which is the given plane. Using the point-to-plane distance formula, we can find the distance from the vertex to the plane to be sqrt(11)/2. Therefore, p = sqrt(11)/2. Plugging in the values, we get (x-0)^2 = 2sqrt(11)(y-0), which simplifies to x^2= 2sqrt(11)y. The correct answer is therefore (B).

Find an Equation for the Surface Consisting of All Points