Question

An object is placed beyond the principal focus at a distance of $9\text{cm}$ from it in front of a concave mirror on the principal axis. The image is formed beyond the principal focus at a distance of $16\text{cm}$ from the principal focus. The focal length of the mirror is

• A. $12\text{cm}$
• B. $11\text{cm}$
• C. $10\text{cm}$
• D. $13\text{cm}$

Answer 1

The correct option is A $12\text{cm}$

Let us draw the object and image position.

Here,
Object distance, $u=-(f+x_1)$
Image distance, $v=-(f+x_2)$
Focal length $=f$
From mirror formula,
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$
$\Rightarrow \frac{-1}{(f+x_2)}-\frac{1}{(f+x_1)}=\frac{-1}{f}$
$\Rightarrow f=\frac{(f+x_1)(f+x_2)}{(f+x_1)+(f+x_2)}$
$\Rightarrow f^2+f{x}_1+f^2+f{x}_2=f{x}_2+x_1{x}_2+f{x}_1+f^2$
$\Rightarrow f^2=x_1{x}_2$
$\Rightarrow f=\sqrt{9\times 16}=\pm 12\text{cm}$
As the mirror is concave, $f=-12\text{cm}$
Hence, the focal length of the mirror is $12\text{cm}$