Question

A point object is placed at a distance $12\text{cm}$ from a convex lens of focal length $10\text{cm}$. On the other side of the lens, a convex mirror is placed at a distance of $10\text{cm}$ from the lens such that the image formed by the combination coincides with the object itself. The focal length of the convex mirror is

• A. $20\text{cm}$
• B. $25\text{cm}$
• C. $30\text{cm}$
• D. $35\text{cm}$

Answer 1

The correct option is B $25\text{cm}$

The arrangement of the given combination is shown below.


Let the object be at point $O$.

If the image to be formed at the object itself, this means that these light rays falling on the mirror-lens combination must retrace their path. That will only happen if the light rays are falling normally on the mirror i.e, light ray must pass through the centre of curvature.

For the lens,

$u=-12\text{cm};f=10\text{cm}$

Using lens formula
$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\Rightarrow \frac{1}{v}=\frac{1}{f}+\frac{1}{u}$

$\Rightarrow \frac{1}{v}=\frac{1}{10}-\frac{1}{12}=\frac{6-5}{60}=\frac{1}{60}$

$\Rightarrow v=60\text{cm}$

Since, image should be formed at the centre of curvature of the mirror.

So, $v=10\text{cm}+2f$

$\Rightarrow 10+2f=60\Rightarrow 2f=60-10=50\text{cm}$

$\Rightarrow f=25\text{cm}$

Therefore, $\left(b\right)$ is correct choice,