Question

The relation among \[u\], $ v $ and $ f $ for a mirror is –
A) $ f = uv(u + v) $
B) $ v = fu(u + f) $
C) $ u = fv(f + v) $
D) None of these

Answer 1

Hint: Recall the mirror formula i.e. the relation between object distance \[u\], image distance $ v $ and focal length $ f $ of a mirror -
$ \dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v} $
Now simplify this expression individually to calculate the value of \[u\], $ v $ and $ f $.

Complete step by step answer:
Step1:
The mirror formula for a mirror is given by $ \dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v} $
Where \[u\] is the object distance, $ v $ is the image distance and $ f $ is the focal length of the mirror.
Now calculate the value of \[u\] from above in terms of $ v $ and $ f $. Therefore,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v} $
$\dfrac{1}{u} = \dfrac{1}{f} - \dfrac{1}{v}$
$\dfrac{1}{u} = \dfrac{{v - f}}{{fv}} $
$ \Rightarrow u = \dfrac{{fv}}{{v - f}} $ …………….(1)

Step2:
Now calculate the value of \[v\] from above in terms of $ u $ and $ f $. Therefore,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v} $
$\dfrac{1}{v} = \dfrac{1}{f} - \dfrac{1}{u}$
$\dfrac{1}{v} = \dfrac{{u - f}}{{fu}} $
$ \Rightarrow v = \dfrac{{fu}}{{u - f}} $ …………….(2)

Step3:
Now calculate the value of \[f\] from above in terms of $ u $ and $ v $. Therefore,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$
$\dfrac{1}{f} = \dfrac{{v + u}}{{uv}}$
$ \Rightarrow f = \dfrac{{uv}}{{v + u}} $ ………………(3)

From (1), (2), and (3) we have seen that it does not match with any of the given options. Hence the correct answer is an option (D) i.e. None of these.

Additional Information:
Mirror formula is defined as the relationship between the object distance $ u $, image distance $ v $ and focal length \[f\] of the mirror from the pole of the mirror and this mirror formula is applicable for all the types of mirrors i.e. for spherical as well as for plane mirror.

Note:
Keep in mind that whenever we apply mirror formula in any problem always use the sign conventions which are as follows –
All the distances should be taken from the pole of the mirror.
All the distances measured in the same direction to that of incident light should be taken as positive whereas all the distance measured in the opposite direction to that of incident light should be taken as negative.
All the distances which are vertically up and perpendicular to the principal axis should be taken as positive whereas all the distances which are vertically downward and perpendicular to the principal axis should be taken as negative.