Question

A candle flame 3cm is placed at a distance of 3m from a wall. How far down the wall must a concave mirror be placed in order that it may form an image of flame $9cm$ high on the wall?
A. $225cm$
B. $300cm$
C. $450cm$
D. $650cm$

Answer 1

Hint: Magnification of a lens is the ratio of image height to object height. And it is also equal to the ratio of image distance to the object distance.

Complete step by step answer:
Let us say that the candle is placed at a distance, ${D_0}$ from the mirror.
Let the distance between the mirror and the wall, that is, the image distance is given by ${D_i}$
Since, the candle will be placed between the mirror and the wall, the image distance will be the sum of the distance, between mirror and candle, and the wall and candle.
Then, ${D_i} = 3 + {D_0}$ . . . (1)
The height of the candle flame is ${h_0} = 3cm$
The height of the image of flame is ${h_i} = 9cm$
Magnification of the image is given by,
$m = \dfrac{{{h_i}}}{{{h_o}}}$
$ \Rightarrow m = \dfrac{9}{3} = 3$
But magnification can also be given as
$m = \dfrac{{{D_i}}}{{{D_o}}}$
Therefore, from equation (1), we can write
$m = \dfrac{{3 + {D_0}}}{{{D_0}}}$
By substituting the value of $m$ in the above equation, we get
$3 = \dfrac{{3 + {D_0}}}{{{D_0}}}$
$ \Rightarrow 3 = \dfrac{3}{{{D_0}}} + 1$
$ \Rightarrow 2 = \dfrac{3}{{{D_0}}}$
Rearranging it we get
\[{D_0} = \dfrac{3}{2}m\]
$ \Rightarrow {D_0} = 1.5m$
Therefore, the distance between the wall and the mirror is given by
$1.5 + 3 = 4.5m$
Since, the options are in centimeter, we will convert it into centimeter.
Thus the distance between the wall and the mirror will be
$4.5 \times 100 = 450cm$

Therefore, form the above explanation the correct option is (C) $450cm$.

Note:In this question, you need to be able to visualize the question properly. You need to understand that the candle will be placed between the mirror and the wall so that the reflection of the candle from the mirror could fall on the wall.