Question

Find the mirror image of the point $(1,2,2)$ on the line
$\frac{x-5}{2}=\frac{y-3}{2}=\frac{z-2}{1}$ .

• A. $(23,40,38)$
• B. $(23,-40,-38)$
• C. $\left(\frac{23}{9},-\frac{40}{9},-\frac{38}{9}\right)$
• D. $\left(\frac{41}{9},-\frac{4}{9},-\frac{2}{9}\right)$

Answer 1

The correct option is D $\left(\frac{41}{9},-\frac{4}{9},-\frac{2}{9}\right)$

Coordinates of any point on the given line are $(2t+5,2t+3,t+2)$
Let this point be the foot of the perpendicular from the point $(1,2,2)$ then direction ratios of the perpendicular is $2t+4,2t+1,t$
Since both are perpendicular, so
$2\times \left(2t+4\right)+2\times \left(2t+1\right)+t=09t+10=0t=-\frac{10}{9}$
The coordinate of foot of the perpendicular is$\left(\frac{25}{9},\frac{7}{9},\frac{8}{9}\right)$
This point is the midpoint of $(1,2,2)$ and mirror of $(1,2,2)$
So the coordinate of mirror of $(1,2,2)$ is $\left(2\times \frac{25}{9}-1,2\times \frac{7}{9}-2,2\times \frac{8}{9}-2\right)=\left(\frac{41}{9},-\frac{4}{9},-\frac{2}{9}\right)$