QUESTION 10.3
Writing the Equation of a Sphere Given its Center and Radius Choose one $\cdot 5$ points
Write the equation of a sphere with center $\vec{c}=(2,1,-5)$ and radius $R=3$
\[
(x-2)^{2}+(y-1)^{2}+(z+5)^{2}=9
\]
\[
(z-2)^{2}+(y-1)^{2}+(z+5)^{2}=3
\]
\[
(x+2)^{2}+(y+1)^{2}+(z-5)^{2}=9
\]

Step 1 ：The equation of a sphere in 3D space is given by the formula: \((x-h)^{2}+(y-k)^{2}+(z-l)^{2}=r^{2}\) where (h, k, l) are the coordinates of the center of the sphere and r is the radius of the sphere.

Step 2 ：In this case, the center of the sphere is given as (2, 1, -5) and the radius is given as 3.

Step 3 ：Therefore, the equation of the sphere should be: \((x-2)^{2}+(y-1)^{2}+(z+5)^{2}=3^{2}\)

Step 4 ：The equation of the sphere is correctly calculated as \((x-2)^{2}+(y-1)^{2}+(z+5)^{2}=9\). This matches the first option provided in the question.

Step 5 ：Final Answer: \(\boxed{(x-2)^{2}+(y-1)^{2}+(z+5)^{2}=9}\)