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, what is the size of image?

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

Answer 1

Answer: (C)

Using the relation for the focal length of concave mirror
$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}\,\,\,\,...(1)$
Differentiating equation (1), we obtain
$0=-\frac{1}{v^{2}} d v-\frac{1}{u^{2}} d u$
So, $d v=-\frac{v^{2}}{u^{2}} \times b\,\,\,...(2)$
(Here: $d u=b$ )
From equation (1)
$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{u-f}{f u}$
or $\frac{u}{v}=\frac{u-f}{f} $
$\frac{v}{u}=\frac{f}{u-f} \,\,\,\,...(3)$
Now, from equations (2) and (3), we get
$d v=-\left(\frac{f}{u-f}\right)^{2} b$
Therefore, size or image is $=\left(\frac{f}{u-f}\right)^{2} b$