The plane which bisects the line joining the points
Question:

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

1. $(4,0,-1)$

2. $(4,0,1)$

3. $(0,1,-1)$

4. $(0,-1,1)$

Correct Option: 1

Solution:

$\mathrm{PA}=\mathrm{PB}$

$\Rightarrow \quad \mathrm{PA}^{2}=\mathrm{PB}^{2}$

$\Rightarrow \quad(\alpha-4)^{2}+(\beta+2)^{2}+(\gamma-3)^{2}$

$=(\alpha-2)^{2}+(\beta-4)^{2}+(\gamma+1)^{2}$

$\Rightarrow \quad-4 \alpha+12 \beta-8 \gamma=-8$

$\Rightarrow \quad 2 x-6 y+4 z=4$