Let $a, b \in R$. If the mirror image of the point $\mathrm{p}(\mathrm{a}, 6,9)$ w.r.t. line…
Let $a, b \in R$. If the mirror image of the point $\mathrm{p}(\mathrm{a}, 6,9)$ w.r.t. line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20, b,-a-9)$, then $|a+b|$ is equal to
88
86
90
84
Solution
Note that mid-point of the line joining points $(a, 6,9)$ and $(20, b,-a-9)$ lies on the given line.
$\therefore \quad$ The midpoint is $\left(\frac{\mathrm{a}+20}{2}, \frac{6+\mathrm{b}}{2}, \frac{9-\mathrm{a}-9}{2}\right)$
Substituting this point in the equation of the given line, we get
$\begin{array}{ll}
& \frac{\frac{a+20}{2}-3}{7}=\frac{\frac{6+b}{2}-2}{5}=\frac{\frac{-a}{2}-1}{-9} \\
\therefore \quad & \frac{a+14}{14}=\frac{a+2}{18} \\
\therefore \quad & a=-56 \\
\therefore \quad & \frac{-56+4}{14}=\frac{2+b}{10} \\
\therefore \quad & b=-32 \\
\therefore \quad & |a+b|=88
\end{array}$