Answer
a) $a^{2}=(z-a)^{2}+(x-a)^{2}$
b) $a^{2}=(y-a)^{2}+(x-a)^{2}$
c) $a^{2}=(z-a)^{2}+(y-a)^{2}$
Work Step by Step
a) In this case the cylinder is parallel to the coordinate planes $x y$ and $y z$ Thus, its equation does not depended on $y .$ The projection on the $x z$ plane is a circle centered at $(a, 0, a)$ of radius $a=r$. Thus, the equation of the cylinder can be written as
\[
a^{2}=(z-a)^{2}+(x-a)^{2}
\]
b) In this case, the cylinder is parallel to the planes $ x z $ and $ y z $ and its projection on the $x y$ plane is the circle centered at $C(a, a, 0)$ of radius $a=r$. Therefore, the equation of the cylinder can be written as
\[
a^{2}=(y-a)^{2}+(x-a)^{2}
\]
c)The cylinder in this case is parallel to the plane $ x y $ and $ x z $. Since its projection on the $y z$ plane is a circle centered at $C(0, a, a)$ of radius $r=a$, its equation can be written as
\[
a^{2}=(z-a)^{2}+(y-a)^{2}
\]
