A converging mirror M_1, a point source S and a diverging mirror … - Vidyadip

Question

A converging mirror $M_1$, a point source $S$ and a diverging mirror $M_2$ 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 $M_2$.

✓

Answer

As shown in figure, for $1^{st}$ reflection in $M_1, u = -30cm, f = -20cm$

$\Rightarrow\frac{1}{\text{v}}+\frac{1}{-30}=-\frac{1}{20}\Rightarrow\text{v}=-60\text{cm}.$
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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