Q. 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) $\sqrt{x_{1}x_{2}}$
Explanation
(d) $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$
$\frac{1}{f - x_{1}} + \frac{1}{f - x_{2}} = \frac{1}{f}$
Or $\ \frac{f - x_{2} + f - x_{1}}{\left( f - x_{1} \right)(f - x_{2})} = \frac{1}{f}$
Or $f^{2} - fx_{2} - fx_{1} + x_{1}x_{2} = 2f^{2} - f(x_{1} + x_{2})$
Or $f^{2} = x_{1}x_{2}\ $or$\ f = \sqrt{x_{1}x_{2}}$
This is Newton’s mirror formula
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.