Question

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

• A. $x_1{x}_2$
• B. $\frac{x_1+x_2}{2}$
• C. $\sqrt{\frac{x_1}{x_2}}$
• D. $\sqrt{x_1{x}_2}$

Answer 1

The correct option is D $\sqrt{x_1{x}_2}$

Given,

The image distance from the focus is given as $x_2$, and object distance from the focus is $x_1$, hence

$v=f-x_2$ and $v=f-x_1$

Now using the mirror formula

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}\Rightarrow \frac{1}{f}=\frac{1}{f-x_2}+\frac{1}{f-x_1}$

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

$f\left(2f-x_1-x_2\right)=\left(f-x_1\right)\left(f-x_2\right)$

$2f^2-f{x}_1-f{x}_2=f^2-f{x}_1-f{x}_2+x_1{x}_2$

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

Therefore the focal length is $f=\sqrt{x_1{x}_2}$ .