Question

The reflection of the point $P(1,0,0)$ in the line $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ is

• A. $(3,-4,-2)$
• B. $(5,-8,-4)$
• C. $(1,-1,-10)$
• D. $(2,-3,8)$

Answer 1

The correct option is B $(5,-8,-4)$

Coordinates of any point $Q$ on the given line are $(2r+1,-3r-1,8r-10).$

So the direction ratios of $PQ$ are $2r,-3r-1,8r-10.$

Now $PQ$ is perpendicular to the given line.

if $2\left(2r\right)-3(-3r-1)+8(8r-10)=0$

$\Rightarrow 77r-77=0\Rightarrow r=1$

and the coordinates of $Q,$ the foot of the perpendicular from $P$ on the line are $(3,-4,-2).$

Let $R(a,b,c)$ be the refelction of $P$ in the given lines then $Q$ is the mid-point of $PR$

$\Rightarrow \frac{a+1}{2}=3,\frac{b}{2}=-4,\frac{c}{2}=-2$

$\Rightarrow a=5,b=-8,c=-4$

and the coordinate of the required point are $(5,-8,-4)$