Question

An object is placed at a distance of $15cm$ from a convex lens of focal length $10cm$. On the other side of the lens, a convex mirror is placed at its focus such that the image formed by the combination coincides with the object itself. The focal length of the convex mirror is (in $cm$ )

Answer 1

Answer: 10.00

For lens, using lens maker's formula
$\frac{1}{f}=\left(\frac{{\mu }_2}{{\mu }_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
$\frac{1}{10}=\left(\frac{1.5}{1}-1\right)\left(\frac{1}{R}-\frac{1}{-R}\right)$
$\Rightarrow \frac{1}{10}=0.5\times \frac{2}{R}$
$\Rightarrow R=10cm$

Focal length of equivalent mirror:
$\frac{1}{f_{\text{eq }}}=\frac{1}{10}+\frac{1}{5}+\frac{1}{10}=\frac{2}{5}$
$f_{\text{eq }}=5/2cm$
So equivalent mirror is a concave mirror and object must be placed at center of curvature of equivalent mirror, i. e. at distance $5cm$ from its pole, so that its image coincides with it.