A converging mirror $\mathrm{M}_1$, a point source S and a diverging mirror $\mathrm{M}_2$ are arranged as shown in figure. The source is placed at a distance of 30 cm from $\mathrm{M}_1$. 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.
- Find the distance between the two mirrors.
- Find the location of the image formed by the single reflection from $\mathrm{M}_2$.
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- As shown in figure, for $1^{st}$ reflection in $M_1, u = -30cm, f = -20cm$
So, for 2^{nd} reflection in $M_2$
$u = 60 - (30 + x) = 30 - x$
$v = -x; f = 20cm$
$\Rightarrow\frac{1}{30-\text{x}}-\frac{1}{\text{x}}=\frac{1}{20}$
$\Rightarrow \frac{\text{x}-30+\text{x}}{\text{x}(30-\text{x})}=\frac{1}{20}$
$\Rightarrow40\text{x}-600=30\text{x}-\text{x}^2$
$\Rightarrow\text{x}^2+10\text{x}-600=0$
$\Rightarrow\text{x}=\frac{10\pm50}{2}=\frac{40}{2}=20\text{cm}$ or $-30\text{cm}$
$\therefore$ Total distance between the two lines is $20 + 30 = 50cm.$
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