The equation of the plane bisecting the line segment joining the points $(2,0,6)$ and $(-6,2,4)$ and perpendicular to it, is

Let variable point on the plane is $(x, y, z)$. Mid-point of the line segment joining the points $(2,0,6)$ and $(-6,2,4)=(-2,1,5)$ Direction ratio of the line segment joining the points $(2,0,6)$ and $(-6,2,4)$ $$ \Rightarrow(-6-2),(2-0),(4-6)-8,2,-2=a_1, b_1, c_1 $$ direction ratios of line joining $(x, y, z)$ and $(-2,1,5)$ $$ =(x+2),(y-1),(z-5)=a_2, b_2, c_2 $$ direction ratio's $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are perpendicular so, $$ a_1 a_2+b_1 b_2+c_1 c_2=0 $$ So, required equation of plane is $$ \begin{array}{rrrl} \Rightarrow & -8(x+2)+2(y-1)-2(z-5) & =0 \\ \Rightarrow & -8 x-16+2 y-2-2 z+10 & =0 \\ \Rightarrow & -8 x+2 y-2 z-8 & =0 \\ \Rightarrow & & 4 x-y+z+4 & =0 . \end{array} $$