Question

A short linear object of length b lies along the axis of a concave mirror of focal length f at a distance u from the pole of the mirror. the size of the length is

• A. $\left(\frac{f}{u-f}\right)b$
• B. $\left(\frac{f}{u-f}\right)b^2$
• C. ${\left(\frac{f}{u-f}\right)}^{2}b$
• D. $\left(\frac{f}{u-f}\right)$

Answer 1

Answer: (C)

${\left(\frac{f}{u-f}\right)}^{2}b$

The mirror formula is given as

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$----------$\left(1\right)$

$0=-\frac{dv}{v^2}-\frac{du}{u^2}$

$\Rightarrow \frac{dv}{du}=-\frac{v^2}{u^2}=-{\left(\frac{v}{u}\right)}^2$----------$\left(2\right)$

From equation $\left(1\right)$

$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{u-f}{fu}$

$\Rightarrow \frac{v}{u}=\frac{f}{u-f}$

Substituting this in equation $\left(2\right)$

$\frac{dv}{du}=-{\left(\frac{f}{u-f}\right)}^2$

We have $du=b$

$\therefore dv={\left(\frac{f}{u-f}\right)}^{2}b$

Hence the correct answer is option (C).