Question

A converging lens of focal length $15cm$ and a converging mirror of focal length $10cm$ are placed $50cm$ apart with common principal axis. A point source is placed in between the lens and the mirror at a distance of $40cm$ from the lens. Where should a point source be placed on the principal axis so that the two images form at the same place?

Answer 1

Let the source be placed at a distance $x$ from the lens as shown, so that images formed by both coincide.

For the lens, $\frac{1}{v_l}-\frac{1}{-x}=\frac{1}{15}\Rightarrow v_l=\frac{15x}{x-15}$............(1)

Fro the mirror, $u=-(50-x),f=-10cm$

So, $\frac{1}{v_m}+\frac{1}{-(50-x)}=-\frac{1}{10}$

$\Rightarrow \frac{1}{v_m}=\frac{1}{-(50-x)}-\frac{1}{10}$

So, $v_m=\frac{10(50-x)}{x-40}$...............(2)

Since the lens and mirror are $50cm$ apart,

$v_l-v_m=50\Rightarrow \frac{15x}{x-15}-\frac{10(50-x)}{(x-40)}=50$

$\Rightarrow x=30$ cm

So, the source should be placed $30$ cm from the lens