Let a , b ∈ R . If the mirror image of the point P a , 6 , 9 with respect to the line x - 3 7 = y - 2…

Let a,b∈R. If the mirror image of the point Pa,6,9 with respect to the line x-37=y-25=z-1-9 is 20, b,-a-9, then |a+b| is equal to:
  1. 86
  2. 90
  3. 84
  4. 88

Solution

$P(a,6,9)$ $\frac{x-3}{7} = \frac{y-2}{5} = \frac{z-1}{-9}$ $Q = (20,b,-a-9)$ Mid point of $PQ = \left(\frac{a+20}{2}, \frac{b+6}{2}, -\frac{a}{2}\right)$ lies on line $\frac{\frac{20+a}{2}-3}{7} = \frac{\frac{b+6}{2}-2}{5} = \frac{-\frac{a}{2}-1}{-9}$ $\frac{a+20-6}{14} = \frac{b+6-4}{10} = \frac{-a-2}{-18}$ $\frac{14+a}{14} = \frac{b+2}{10} = \frac{a+2}{18}$ $\frac{a+14}{14} = \frac{a+2}{18}$ $18a+252 = 14a+28$ $4a = -224$ $a = -56$ $\frac{b+2}{10} = \frac{a+2}{18}$ $\frac{b+2}{10} = \frac{-54}{18}$ $\frac{b+2}{10} = -3 \Rightarrow b = -32$ $|a+b| = |-56-32| = 88$

Asked in: JEE Main 2021 (24 Feb Shift 2)

Practice more Line and Plane questions on Aicharya