Question

A line L is passing through points A(1, 3, 2) and B(2, 2, 1). If mirror image of point P(1, 1, -1) in the line L is (x, y, z) then $x + y + z =$

• A. $\frac{10}{3}$
• B. $\frac{13}{3}$
• C. $\frac{14}{3}$
• D. $\frac{23}{3}$

Hint:

Foot of perpendicular $M$ is the average of a point $P$ and its mirror image $P'$.

Answer 1

The Correct Option is B

Step 1: Find Projection of P on line L
Line L: $\vec{r} = (1, 3, 2) + \lambda(1, -1, -1)$.
Let $M$ be the foot of perpendicular. $M = (1+\lambda, 3-\lambda, 2-\lambda)$.
$\vec{PM} \cdot \vec{d} = 0 \implies (\lambda, 2-\lambda, 3-\lambda) \cdot (1, -1, -1) = 0$.
$\lambda - (2-\lambda) - (3-\lambda) = 0 \implies 3\lambda - 5 = 0 \implies \lambda = 5/3$.

Step 2: Find coordinates of M
$M = (8/3, 4/3, 1/3)$.

Step 3: Mirror image point
$M = \frac{P + P'}{2} \implies P' = 2M - P$.
$x = 2(8/3) - 1 = 13/3$
$y = 2(4/3) - 1 = 5/3$
$z = 2(1/3) - (-1) = 5/3$.

Step 4: Sum
$x+y+z = (13+5+5)/3 = 23/3$.
Final Answer: (D)