Question

A point source of light B is placed at a distance L in front of the center of the mirror of width‘d’ hung vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror at a distance 2L from it as shown in the fig. The greatest distance over which he can see the image of the light source in the mirror is:
(A).$\dfrac{d}{2}$
(B).$d$
(C). $2d$
(D).$3d$

Answer 1

As the man goes further and further away from the mirror, the light rays being reflected from the mirror to his eyes decreases until he reaches a distance when no light rays from the mirror reach him. The light rays being reflected from the furthest corners of the mirror will give the greatest distance. Using triangle similarity and laws of reflection, the greatest distance can be calculated.

Complete answer:
A plane mirror makes a virtual and erect image at the back of the mirror. The image is formed at the same distance as the object from the mirror.
Reflection is governed by two laws which states that-
The angle of reflection is equal to the angle of incidence
The same plane contains the incident ray, reflected ray and the normal.
Using the following laws of reflection we can deduce over what distance the image of light will be visible to the man.

In $\Delta ABC\,\text{and}\,\Delta ADC$
$\angle i=\angle r$ (by laws of reflection)
$\angle DCA=\angle BCA$
Therefore by AA similarity
$\Delta ABC\sim \Delta DBC$
Therefore by spst, we have,
$\dfrac{BC}{BA}=\dfrac{DC}{CA}$
$\Rightarrow BC=DC=\dfrac{d}{2}$ - (1)
[ B is hung in the centre of mirror]
In
$\text{ }\\!\\!\Delta\\!\\!\text{ DAC}\,\text{and}\,\text{ }\\!\\!\Delta\\!\\!\text{ PAQ}$
$\angle PAQ=\angle DAC$ (Coinciding angles)
$\angle DCA=\angle PQA$ (${{90}^{o}}$each)
By AA similarity,
$\Delta DAC\sim \Delta PAQ$
By spst, we have,
$\dfrac{PQ}{DC}=\dfrac{AQ}{AB}$
$\begin{aligned} & \Rightarrow \dfrac{PQ}{\dfrac{d}{2}}=\dfrac{2L}{L} \\\ & PQ=2\times \dfrac{d}{2} \\\ \end{aligned}$
$\therefore PQ=d$
The greatest distance over which the image is still visible to the man is $d$.

Hence, the correct option is (B).

Note:
For an image to be formed in a mirror, it should have a polished and shiny surface as image is formed by the phenomenon of reflection. The plane mirror is laterally inverted. The other types of mirrors are concave and convex mirrors. They can form real as well as virtual images for different positions of the object.