Find the equation of the set of points M(x,y,z) at distance…

Find the equation of the set of points $M(x,y,z)$ at distance $2$ from $C(1,-1,2)$.

Explanation of Method

The set of points at a fixed distance from a given point is a sphere.

\[ |MC|=2\iff\sqrt{(x-1)^2+(y+1)^2+(z-2)^2}=2 . \]

Both sides are nonnegative, so squaring is an equivalence: $(x-1)^2+(y+1)^2+(z-2)^2=4$.

This is the sphere with center $C(1,-1,2)$ and radius $2$. Expanded: $x^2+y^2+z^2-2x+2y-4z+2=0$.

$C(1,-1,2)$ $M$ $2$

All points $M$ at distance $2$ from $C$: the sphere $(x-1)^2+(y+1)^2+(z-2)^2=4$.

\[\boxed{\,(x-1)^2+(y+1)^2+(z-2)^2=4\,}\]