Find the standard form of the equation of the sphere with the given characteristics.
Endpoints of a diameter: $( - 8,4 , - 2 ) , ( - 8 , - 2,6 )$
A) $( x + 16 ) ^ { 2 } + ( x - 2 ) ^ { 2 } + ( x - 4 ) ^ { 2 } = 25$
B) $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 50$
C) $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 100$
D) $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 25$
E) $( x - 8 ) ^ { 2 } + ( x + 1 ) ^ { 2 } + ( x + 2 ) ^ { 2 } = 25$

Question

Quizplus · Accepted Answer

The center of the sphere is the midpoint of the given endpoints of the diameter, which is $(-8, 1, 2)$.
The radius of the sphere is half the distance between the endpoints of the diameter. Using the distance formula, we have:
\begin{align*}
d &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \
&= \sqrt{(-8 + 8)^2 + (4 + 2)^2 + (-2 - 6)^2} \
&= \sqrt{64 + 36 + 64} \
&= 6\sqrt{6}
\end{align*}
Therefore, the standard form of the equation of the sphere is:
$$(x+8)^2 + (y-1)^2 + (z-2)^2 = 6^2 \cdot 6$$
Simplifying this gives:
$$(x+8)^2 + (y-1)^2 + (z-2)^2 = 216$$
This matches choice D.

Find the Standard Form of the Equation of the Sphere (−8,4,−2),(−8,−2,6)( - 8,4 , - 2 ) , ( - 8 , - 2,6 )(−8,4,−2),(−8,−2,6)

Find the Standard Form of the Equation of the Sphere $( - 8,4 , - 2 ) , ( - 8 , - 2,6 )$