Question

Let the mirror image of the point (1, 3, a) with respect to the plane r · (2 i - j + k) - b = 0 be (-3, 5, 2). Then, the value of |a + b| is equal to ________.

Hint:

For mirror images, two things are always true: the midpoint lies on the plane, and the line joining the points is perpendicular to the plane.

Answer 1

Correct Answer: 1

Step 1: The midpoint of point $P(1, 3, a)$ and image $P'(-3, 5, 2)$ lies on the plane. Midpoint $M = (\frac{1-3}{2}, \frac{3+5}{2}, \frac{a+2}{2}) = (-1, 4, \frac{a+2}{2})$.
Step 2: $M$ satisfies $2x - y + z - b = 0$: $2(-1) - 4 + \frac{a+2}{2} - b = 0 \implies -6 + \frac{a+2}{2} = b$.
Step 3: The vector $PP'$ is parallel to the normal $(2, -1, 1)$. $\vec{PP'} = (-4, 2, 2-a)$. Comparing ratios: $\frac{-4}{2} = \frac{2}{-1} = \frac{2-a}{1} \implies -2 = 2-a \implies a = 4$.
Step 4: Substitute $a=4$ into Step 2: $-6 + \frac{6}{2} = b \implies b = -3$.
Step 5: $|a + b| = |4 - 3| = 1$.