Question

The plane which bisects the line segment joining the points $(-3,-3,4)$ and $(3,7,6)$ at right angles, passes through which one of the following points?

• A. $(4,-1,7)$
• B. $(4,1,-2)$
• C. $(2,1,3)$
• D. $(-2,3,5)$

Answer 1

The correct option is B $(4,1,-2)$

Let $M$ is the mid-point of $AB$ and lies on the plane

$\therefore M(0,2,5)$

$\because$ The required plane is perpendicular bisector of line joining $A,B,$ so direction ratios of normal to the plane is proportional to the direction ratios of line joining $A,B.$
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=6\hat{i}+10\hat{j}+2\hat{k}$

So, direction ratios of normal to the plane are $6,10,2.$
Hence, normal vector to the plane $\overrightarrow{n}=3\hat{i}+5\hat{j}+\hat{k}$
and $\overrightarrow{a}=2\hat{j}+5\hat{k}$
The equation of plane is
$(\overrightarrow{r}-\overrightarrow{a})\cdot \overrightarrow{n}=0$
$\Rightarrow 3(x-0)+5(y-2)+1(z-5)=0$
$\Rightarrow 3x+5y+z-15=0$
Clearly, $(4,1,-2)$ satisfies the above equation.