Chapter 14.1, Problem 49E

To determine

(a)

To show that the upper and lower spheres have centers $C_1=\left(0,0,r\sqrt{a^2+b^2+1}\right)$ and $C_2=\left(0,0,-r\sqrt{a^2+b^2+1}\right)$.

To determine

(b)

To show that the points where the plane is tangent to the sphere are $F_1=\frac{r}{\sqrt{a^2+b^2+1}}\left(a,b,a^2+b^2\right)$ and $F_2=\frac{-r}{\sqrt{a^2+b^2+1}}\left(a,b,a^2+b^2\right)$.

To determine

(c)

To verify that $K=2r\sqrt{1+a^2+b^2}$.