Answer
$(x-\frac{3}{2})^2 + (y-\frac{3}{2})^2 + (z-7)^2 = \frac{13}{2}$
Work Step by Step
Midpoint: $(\frac{1+2}{2},\frac{0+3}{2},\frac{5+9}{2}) = (\frac{3}{2},\frac{3}{2},7)$
Distance between $P$ and $Q$: $\sqrt {(2-1)^2+(3-0)^2+(9-5)^2} = \sqrt {1 + 9 + 16} = \sqrt {26}$
Center = Midpoint
Radius = $\frac{(distance\ between\ P\ and\ Q)}{2}$
Equation of Sphere with Center $(a,b,c)$ and radius $r$
$(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$
Plug in the center and radius:
$(x-\frac{3}{2})^2 + (y-\frac{3}{2})^2 + (z-7)^2 = (\frac{\sqrt {26}}{2})^2$
Simplify:
$(x-\frac{3}{2})^2 + (y-\frac{3}{2})^2 + (z-7)^2 = \frac{13}{2}$
