Mathematics - Vector Question with Solution | TestHub

$P\left(a,6,9\right)$

$\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$

$Q=\left(20,b,-a-9\right)$

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

lies on line

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

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

$\frac{14+a}{14}=\frac{b+2}{10}=\frac{a+2}{18}$

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

$18a+252=14a+28$

$4a=-224$

$a=-56$

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

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

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

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