With a concave mirror, an object is placed at a distance $x_1$ from the principal focus away from the mirror, on the principal axis. The image is formed at a distance $x_2$ from the principal focus also away from the mirror. The focal length of the mirror is

The correct option is D $\sqrt{x_1{x}_2}$
Given that an object is placed at a distance $x_1$ from the principal focus away from the mirror, on the principal axis.
$u=-(f+x_1)$

The image is formed at a distance $x_2$ from the principal focus
$u=-(f+x_2)$
Now from mirror formula

$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$

$\frac{1}{-(f+x_1)}+\frac{1}{-(f+x_2)}=\frac{1}{-f}$

$\frac{1}{f+x_1}+\frac{1}{f+x_2}=\frac{1}{f}$

$\frac{f+x_2+f+x_1}{(f+x_1)(f+x_2)}=\frac{1}{f}$

$\Rightarrow f^2+f{x}_2+f{x}_1+x_1{x}_2=2f^2+f(x_1+x_2)$

$\Rightarrow f^2=x_1{x}_2\Rightarrow f=\sqrt{x_1{x}_2}$

This is Newton's mirror formula.

Why this question? Tip: If the distance of object and image are not measured from optical centre (C), but from first and second principal foci respectively, then newton’s formula can be used.