Question

With a concave mirror, an object is placed at a distance $X_1$ from the principle focus, on the principle axis. The image is formed at a distance $X_2$ from the principle focus. The focus length of the mirror is

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

Answer 1

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

$\begin{array}{c}\text{Here}u=(f+x_1)\\ v=-(f+x_2)\\ \text{from mirror formula}\\ \begin{array}{cc}-\frac{1}{f}& =\frac{-1}{(f(+x_2)}+\frac{-1}{(f+x_1)}\\ \frac{1}{f}& =\frac{f+x_1+f+x_2}{(f+x_2)(f+x_1)}\end{array}\end{array}$

$\begin{array}{c}\Rightarrow 2f^2+[(x_1+x_2)=f^2+f{\mu }_1+f{x}_2+x_1{x}_2\\ f^2=x_1{x}_2\\ f=\sqrt{x_1{x}_2}\end{array}$