Question

A person $AB$ of height $170\text{cm}$ is standing in front of a plane mirror. His eyes are at height $164\text{cm}$. At what distance from $P$ should a hole be made in the mirror so that he cannot see his hairs?

• A. $165\text{cm}$
• B. $167\text{cm}$
• C. $161\text{cm}$
• D. $163\text{cm}$

Answer 1

The correct option is B $167\text{cm}$

To see the image of hair, the rays incident from hair must reach the eye of the person after reflection.
Let the hole be made at a point $M$ on the mirror where light coming from hair is reflected.


From the geometry of the diagram,
$PM=BC$
$\Rightarrow PM=BE+CE$
$\Rightarrow PM=BE+\left(\frac{AE}{2}\right)$
$[\because$ In $\Delta AEM,AM=ME$ $($ Isosceles triangle $)$ and $MC$ is the perpendicular bisector of $AE]$
$\Rightarrow PM=164+\left(\frac{6}{2}\right)$
$\therefore PM=167\text{cm}$
Hence, option $\left(a\right)$ is the correct answer.