$\text{ The plane which bisects the line joining the points }(4,-2,3)\text{ and }(2,4,-1)\text{ at right angles also passes through }$the point :

$\text{ Midpoint of the line segment joining the points }(4,-2,3)\text{ and }(2,4,-1)\text{ is }(3,1,1)\text{ and direction ratios of the line }$

$\text{ joining the points }(4,-2,3)\text{ and }(2,4,-1)\text{ are }⟨2,-6,4⟩=⟨1,-3,2⟩$

$\text{ Hence, the equation of the perpendicular bisecting plane of the segment joining }(4,-2,3)\text{ and }(2,4,-1)\text{ is }$

$\begin{array}{c}1\left(x-3\right)-3\left(y-1\right)+2\left(z-1\right)=0\\ x-3y+2z-3+3-2=0\\ x-3y+2z-2=0\end{array}$

\text { It also passes through }(4,0,-1)