Let the mirror image of the point (1,3,a) with respect to the plane r(2 i- j+ k)-b=0 be (-…

Question

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

Answer: 1

Accepted Answer

Plane: $2x-y+z=b$. Midpoint $R=\left(-1,4,\dfrac{a+2}{2}\right)$ lies on the plane:
$$-2-4+\frac{a+2}{2}=b \implies a=2b+10.\quad(*)$$
$\vec{QP}=(4,-2,a-2)$ is parallel to the normal $(2,-1,1)$:
$$\frac{2}{4}=\frac{1}{a-2} \implies a-2=2 \implies a=4.$$
From $(*)$: $b=-3$.
$|a+b|=|4-3|=1.$

Solutions · 1