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. $\left(x_{1}+x_{2}\right) / 2$
• C. $\sqrt{x_{1} / x_{2}}$
• D. $\sqrt{x_{1} x_{2}}$

Answer 1

Answer: (D)

$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$
$\frac{1}{f-x_{1}}+\frac{1}{f-x_{2}}=\frac{1}{f} $
or $ \frac{f-x_{2}+f-x_{1}}{\left(f-x_{1}\right)\left(f-x_{2}\right)}=\frac{1}{f}$
or $f^{2}-f x_{2}-f x_{1}+x_{1} x_{2}=2 f^{2}-f\left(x_{1}+x_{2}\right)$
or $f^{2}=x_{1} x_{2}$ or $f=\sqrt{x_{1} x_{2}}$
This is Newton's mirror formula.