Question

Consider a concave mirror kept at origin with focal length 40 cm. Parallel rays which subtend an angle

=

1

40 radian are incident on it. A convex mirror is kept at a distance 25 cm from the first mirror as

shown. Find the y-coordinate of the image (in cm) formed by the system of mirrors after two reflections.​

Answer 1

Given that,

Image distance= 40 cm

Angle = 40 radian

Distance of convex mirror from first mirror = 25 cm

The incident rays are parallel.

So, the first image is formed at focus of concave mirror

According to figure,

$\tan\theta\approx \theta$

$\theta=\dfrac{1}{40}$

We need to calculate the height of image

Using formula of angle

$\theta=\dfrac{Perpendicular}{Base}$

Put the value into the formula

$\theta=\dfrac{y_{1}}{OA}$

$y_{1}=\theta\times OA$

$y_{1}=\dfrac{1}{40}\times40$

$y_{1}=1\ cm$

For convex mirror.

Object distance = 15 cm

We need to calculate the focal length

Using formula of mirror

$\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}$

Put the value into the formula

$\dfrac{1}{f}=\dfrac{1}{40}-\dfrac{1}{-15}$

$\dfrac{1}{f}=\dfrac{11}{120}$

$f=10\ cm$

We need to calculate the height of image

Using formula of magnification

$m=\dfrac{h_{i}}{h_{o}}=\dfrac{f}{f+u}$

$\dfrac{y_{2}}{y_{1}}=\dfrac{f}{f+u}$

Put the value into the formula

$\dfrac{y_{2}}{1}=\dfrac{10}{10+15}$

$y_{2}=0.4\ cm$

Hence, The y-coordinate of the image formed by the system of mirrors after two reflections is 0.4 cm.