Find the sphere of radius R=a (a> 0) tangent to all three…

Find the sphere of radius $R=a$ ($a\gt 0$) tangent to all three coordinate planes.

Step 1: Center

The center $(p,q,r)$ is at distance $|p|$, $|q|$, $|r|$ from the planes $x=0$, $y=0$, $z=0$. Tangency means each of these equals $a$: \[|p|=|q|=|r|=a\ \Longrightarrow\ (p,q,r)=(\pm a,\pm a,\pm a).\]

Step 2: Equations

\[(x-\varepsilon_1a)^2+(y-\varepsilon_2a)^2+(z-\varepsilon_3a)^2=a^2,\qquad \varepsilon_i\in\{1,-1\}\]

\[ x^2+y^2+z^2-2a(\varepsilon_1x+\varepsilon_2y+\varepsilon_3z)+2a^2=0 . \]

There are $8$ spheres, one in each octant. The one in the first octant is $(x-a)^2+(y-a)^2+(z-a)^2=a^2$. It touches the three planes at $(a,a,0)$, $(0,a,a)$ and $(a,0,a)$.