A converging mirror $M_1$ a point source $S$ and a diverging mirror $M_2$ are arranged as shown in below fig. The source is placed at a distance of $30 cm$ from $M_1$. The focal length of each of the mirrors is $20 cm$. Consider only the images formed by a maximum of two reflections. It is found that one image is formed on the source itself. The distance between the mirrors is (in cm)

For mirror $M_1$ 

                              $$\begin{gathered} u=-30 cm, \\ f=-20 cm \end{gathered}$$

By mirror formula, we have $\frac{1}{u}+\frac{1}{v}=\frac{1}{f},$

                 $\frac{1}{-30}+\frac{1}{v}=\frac{1}{-20}$

which on solving gives, $v=-60 cm$

For mirror $M_2$:The image formed by mirror $M_1$ behaves like virtual object for mirror $M_2$. Let it is at a distance $x$ from the pole of mirror $M_2$. Thus

$$\begin{gathered} u_2=+x \\ v_2=-\left(30-x\right) \end{gathered}$$

Again by mirror formula, we have

$\frac{1}{x}+\frac{1}{-\left(30-x\right)}=\frac{1}{20}$

which on solving gives $x= 10 cm \text{or} x=60 cm,$

$x=60 cm$ is not possible, $\therefore x= 10 cm$

Thus the separation between the mirrors $=60-10=50 cm.$