The 3-D Coordinate System — Question 5

Question 5 – Solution

Strategy Tangency to the $xy$-plane makes the radius equal to the center’s height. The condition that the sphere lies above the plane forces $c>0$.

Build the equation Here $r=c$. Because $P$ is on the sphere, $$(5-2)^2+(3+1)^2+(2-c)^2=c^2.$$ Thus $25+(2-c)^2=c^2$, so $29-4c=0$ and $c=29/4$.

Result The radius is $29/4$, and $$\boxed{(x-2)^2+(y+1)^2+\left(z-\frac{29}{4}\right)^2=\left(\frac{29}{4}\right)^2}.$$

Verification The lowest $z$-coordinate is $c-r=0$, so the sphere is tangent to the plane and otherwise lies above it.