Question

Let $a,b\in R.$ If the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{(x-3)}{7}=\frac{(y-2)}{5}=\frac{(z-1)}{(-9)}$ is $(20,b,-a-9)$ , then $|a+b|$ is equal to:

• A. $86$
• B. $88$
• C. $84$
• D. $90$

Answer 1

The correct option is B

$88$

Explanation for the correct option :

We need to find the values of $|a+b|$

Given: the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{(x-3)}{7}=\frac{(y-2)}{5}=\frac{(z-1)}{(-9)}$ is $(20,b,-a-9)$

So, Mid Point of $P,Q$ will lie on the given line hence,

$\begin{array}{rcl}& & P(a,6,9)\\ \frac{x-3}{7}& =& \frac{y-2}{5}=\frac{z-1}{-9}\\ Q& =& (20,b,-a-9)\\ \frac{{\displaystyle \frac{20+a}{2}}-3}{7}& =& \frac{{\displaystyle \frac{b+6}{2}}-2}{5}=\frac{{\displaystyle \frac{-a}{2}}-1}{-9}\end{array}$

hence,

$\begin{array}{rcl}\frac{14+a}{14}& =& \frac{b+2}{10}=\frac{a+2}{18}\\ & \Rightarrow & a=-56,b=-32\\ & \Rightarrow & \mid a+b\mid =88\end{array}$

Hence the correct option is (B).