$\text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. }$
A) $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$
B) $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$
C) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$
D) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$
E) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$

Question

$\text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. }$ 
A) $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$
B) $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$
C) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$
D) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$
E) $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$

Quizplus · Accepted Answer

Since the sphere touches the xy-plane, its center lies on the line perpendicular to the xy-plane passing through the origin. Therefore, the distance from the center to the xy-plane is the radius of the sphere. The distance can be found as the absolute value of the z-coordinate of the center, so the radius is 7. Thus, the equation of the sphere is:

$$(x-7)^2+(y+5)^2+(z-7)^2=7^2$$

Simplifying the left side gives:

$$(x-7)^2+(y+5)^2+(z-7)^2=49$$

Therefore, the answer is D.

Find an equation of the sphere with center (7,−5,7) that touches the xy-plane. \text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. } Find an equation of the sphere with center (7,−5,7) that touches the xy-plane.

$\text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. }$