Question

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)/5 = (z-1)/(-9) is (20, b, -a - 9), then |a + b| is equal to :

• A. 84
• B. 86
• C. 88
• D. 90

Hint:

For mirror image problems, use two facts: 1. The midpoint lies on the line/plane. 2. The vector joining the point and its image is perpendicular to the line's direction.

Answer 1

The Correct Option is C

Step 1: Let $Q$ be the mirror image $(20, b, -a-9)$. The midpoint $M$ of $PQ$ must lie on the line.
Step 2: $M = \left( \frac{a+20}{2}, \frac{6+b}{2}, \frac{9-a-9}{2} \right) = \left( \frac{a+20}{2}, \frac{6+b}{2}, -\frac{a}{2} \right)$.
Step 3: Substitute $M$ into the line equation: $\frac{\frac{a+20}{2} - 3}{7} = \frac{\frac{6+b}{2} - 2}{5} = \frac{-\frac{a}{2} - 1}{-9} \Rightarrow \frac{a+14}{14} = \frac{b+2}{10} = \frac{a+2}{18}$.
Step 4: Solve $\frac{a+14}{14} = \frac{a+2}{18} \Rightarrow 18a + 252 = 14a + 28 \Rightarrow 4a = -224 \Rightarrow a = -56$.
Step 5: Solve $\frac{-56+14}{14} = \frac{b+2}{10} \Rightarrow -3 = \frac{b+2}{10} \Rightarrow b = -32$.
Step 6: $|a+b| = |-56 - 32| = |-88| = 88$.