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 $88$

$P(a,6,9),Q(20,b,-a-9)$
Mid point of $PQ=(\frac{a+20}{2},\frac{b+6}{2},-\frac{a}{2})$ lie on the line.
$\frac{\frac{a+20}{2}-3}{7}=\frac{\frac{b+6}{2}-2}{5}=\frac{-\frac{a}{2}-1}{-9}$
$\Rightarrow \frac{a+20-6}{14}=\frac{b+6-4}{10}=\frac{-a-2}{-18}$
$\Rightarrow \frac{a+14}{14}=\frac{a+2}{18}$
$\Rightarrow 18a+252=14a+28$
$\Rightarrow 4a=-224$
$\Rightarrow a=-56$

$\frac{b+2}{10}=\frac{a+2}{18}$
$\Rightarrow \frac{b+2}{10}=\frac{-54}{18}$
$\Rightarrow \frac{b+2}{10}=-3\Rightarrow b=-32$
$|a+b|=|-56-32|=88$