A concave and a convex mirror of focal lengths $10\text{cm}$ each are placed at a distance $d\text{cm}$ apart. An object $\text{AB}$ is placed in between the mirrors at a distance of $10\text{cm}$ from the convex mirror. If first ray is incident on the convex mirror and then on the concave mirror then find the distance $d$ between both the mirrors, so that final image will be formed at the object's position.

The correct option is D $25\text{cm}$
For convex mirror:

$u=-10\text{cm}$
$f=+10\text{cm}$

Using mirror formula
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$\Rightarrow \frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{1}{10}-\frac{1}{(-10)}$

$\therefore v=5\text{cm}$


For concave mirror

$v=-(d-10)\text{cm}$
$u=-(d+5)\text{cm}$
$f=-10\text{cm}$

Applying the mirror formula,

$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}$

$\Rightarrow \frac{1}{-(d-10)}=\frac{1}{-10}-\frac{1}{(-(d+5))}$

$\Rightarrow \frac{1}{(10-d)}=\frac{1}{d+5}-\frac{1}{10}$

$\Rightarrow \frac{1}{10-d}=\frac{5-d}{10(d+5)}$

$\Rightarrow 10d+50=50-10d-5d+d^2$

$\Rightarrow 25d=d^2$

$\Rightarrow d(d-25)=0$

$d=0$ and $d=25\text{cm}$

Hence, option (d) is the correct answer.

Why this question: To understand the concept of formation of image at the objects postion after multiple refection .