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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

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$\frac{x_{1} + x_{2}}{2}$

$x_{1} x_{2}$

$\sqrt{x_{1} x_{2}}$

$\sqrt{\frac{x_{1}}{x_{2}}}$

answer is D.

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Detailed Solution

$u = X_{1} + f, V = X_{2} + f$

$f = \frac{u V}{u + V} \Rightarrow \frac{(x_{1} + f) (x_{2} + f)}{(x_{1} + f) + (x_{2} + f)}$

$f \Rightarrow \frac{x_{1} x_{2} + x_{1} f + x_{2} f + f^{2}}{x_{1} + x_{2} + 2 f}$

$\Rightarrow x_{1} f + x_{2} f + 2 f^{2} = x_{1} x_{2} + x_{1} f + x_{2} f + f^{2}$

$2 f^{2} - f^{2} = x_{1} x_{2}$

$∴ f = \sqrt{x_{1} x_{2}}$

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