Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point $( 2,4,5 )$ and the plane $y = - 1$.
A) $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } + 15$, an elliptic paraboloid
B) $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 5 = 0$, a sphere
C) $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } + 1 ,$, a hyperbolic paraboloid
D) $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 1 = 0$, a sphere
E) $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 1$, , a hyperbolic paraboloid

Question

Quizplus · Accepted Answer

The correct surface is an elliptic paraboloid. The distance from a point $(x, y, z)$ to the point $(2, 4, 5)$ is equal to the distance from $(x, y, z)$ to the plane $y = -1$. The equation for the distance to the point is $\sqrt{(x-2)^2 + (y-4)^2 + (z-5)^2}$ and the distance to the plane $y = -1$ is $|y + 1|$. Squaring both sides to eliminate the square root gives $(x-2)^2 + (y-4)^2 + (z-5)^2 = (y + 1)^2$, which simplifies to an equation of the form given in option A, representing an elliptic paraboloid.

Find the Equation of the Surface Satisfying the Conditions, and Identify