Question

The perpendicular bisector of a line segment with end points $\left(1,2 , 6\right)$ and $\left(- 3,6 , 2\right)$ passes through $\left(- 6,2 , 4\right)$ and has the equation of the form $\frac{x + 6}{l}=\frac{y - 2}{m}=\frac{z - 4}{n}$ (where $l,m,n$ are integers, $l$ is a prime number and $l>0$ ), then the value of $lmn-\left(l + m + n\right)$ is equal to

• A. $-3$
• B. $-5$
• C. $-7$
• D. $-9$

Answer 1

Answer: (C)

Midpoint of the line segment is $\left(\frac{1 - 3}{2} , \frac{2 + 6}{2} , \frac{6 + 2}{2}\right)\equiv \left(- 1 ,4 , 4\right)$ $$
Parallel vector to the required line $=\left(- 1 + 6\right)\hat{i}+\left(4 - 2\right)\hat{j}+\left(4 - 4\right)\hat{k}$
$=5\hat{i}+2\hat{j}+0\hat{k}$
Hence, the equation of the line is
$\frac{x + 6}{5}-\frac{y - 2}{2}=\frac{z - 4}{0}$
$\Rightarrow l=5,m=2,n=0$
So, $lmn-\left(l + m + n\right)=0-7=-7$