The equation of the sphere centered at (2,1,-2) and tangent to the xy-plane is

A) $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$
B) $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$
C) $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$
D) $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$
E) $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$

Question

The equation of the sphere centered at (2,1,-2) and tangent to the xy-plane is

A) $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$
B) $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$
C) $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 4$
D) $( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$
E) $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 2 ) ^ { 2 } = 4$

Quizplus · Accepted Answer

The equation of a sphere is given by $(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$, where $(h, k, l)$ is the center of the sphere and $r$ is the radius. Since the sphere is centered at $(2, 1, -2)$ and is tangent to the xy-plane, its radius is equal to the absolute value of the z-coordinate of the center, which is $2$. Therefore, the correct equation is $(x - 2)^2 + (y - 1)^2 + (z + 2)^2 = 4$, which matches choice B.

The Equation of the Sphere Centered at (2,1,-2) and Tangent (x+2)2+(y−1)2+(z+2)2=2( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2(x+2)2+(y−1)2+(z+2)2=2

The Equation of the Sphere Centered at (2,1,-2) and Tangent $( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 2 ) ^ { 2 } = 2$