# Prove the following

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:

1. (1) 86

2. (2) 88

3. (3) 84

4. (4) 90

Correct Option: , 2

Solution:

$P(a, 6,9), Q(20, b,-a-9)$

mid point of $\mathrm{PQ}=\left(\frac{\mathrm{a}+20}{2}, \frac{\mathrm{b}+6}{2},-\frac{\mathrm{a}}{2}\right)$

lie on line

$\frac{\frac{a+20}{2}-3}{7}=\frac{\frac{b+6}{2}-2}{5}=\frac{-\frac{a}{2}-1}{-9}$

$\frac{a+20-6}{14}=\frac{b+6-4}{10}=\frac{-a-2}{-18}$

$\frac{\mathrm{a}+14}{14}=\frac{\mathrm{a}+2}{18}$

$18 a+252=14 a+28$

$a=-56$

$\frac{b+2}{10}=\frac{a+2}{18}$

$\frac{b+2}{10}=\frac{-54}{18}$

$\frac{b+2}{10}=-3 \Rightarrow b=-32$

$|a+b|=|-56-32|=88$