Write the equation of the set of points M(x,y,z) equidistant…

Write the equation of the set of points $M(x,y,z)$ equidistant from $A(2,-1,2)$ and $B(0,1,0)$.

Step 1: Solution

\[ (x-2)^2+(y+1)^2+(z-2)^2=x^2+(y-1)^2+z^2 \]

The squares cancel, leaving

\[ -4x+2y-4z+9=-2y+1\ \Longrightarrow\ -4x+4y-4z+8=0 . \]

\[x-y+z-2=0\]

This is the perpendicular bisector plane of $AB$:

• it contains the midpoint $P(1,0,1)$, since $1-0+1-2=0$;
• its normal $\{1,-1,1\}$ is parallel to $\overrightarrow{AB}=\{-2,2,-2\}$.