The 3-D Coordinate System — Question 2

Question 2 – Solution

Strategy The center is the midpoint of the diameter, and the radius is half its length. Points on the $z$-axis have the form $(0,0,z)$.

Center and radius The midpoint is $C=(2,-1,2)$. Also, $$|AB|=\sqrt{8^2+(-4)^2+(-4)^2}=4\sqrt 6,\qquad r=2\sqrt 6.$$ Therefore the sphere is $$\boxed{(x-2)^2+(y+1)^2+(z-2)^2=24}.$$

$z$-axis intersections Set $x=y=0$: $4+1+(z-2)^2=24$, so $(z-2)^2=19$. The intersection points are $\boxed{(0,0,2\pm\sqrt{19})}$.

Verification Both $A$ and $B$ are $2\sqrt 6$ units from $C$, and their midpoint is $C$.