Question

Find the coordinates of image of point object $\text{P}$ in $\text{mm}$ formed after two successive reflections one at concave mirror and the other at convex.

• A. $(+30\text{cm},-14\text{mm})$
• B. $(+30\text{cm},-7\text{mm})$
• C. $(+20\text{cm},-16\text{mm})$
• D. $(+20\text{cm},-14\text{mm})$

Answer 1

The correct option is A $(+30\text{cm},-14\text{mm})$

$\text{Reflection at concave mirror}\left(M_1\right):$

Object distance, $u_1=-20\text{cm}$
Focal distance, $f_1=-15\text{cm}$

Using mirror formula, $$\frac{1}{v_1}+\frac{1}{u_1}=\frac{1}{f_1}$$ We get,

$\frac{1}{v_1}=\frac{1}{-15}+\frac{1}{20}$

$\Rightarrow \frac{1}{v_1}=\frac{5}{-300}$

$\Rightarrow v_1=-60\text{cm}$

Magnification:

$m_1=\frac{-v_1}{u_1}$

$\Rightarrow m_1=\frac{-(-60)}{-20}$

$\Rightarrow m_1=-3$

Since, initially the height of the point $\text{P}$ is $h=2\text{mm}$, so the height at which the image point will be

$h^{\prime }=|m_1|\times h$

$\Rightarrow h^{\prime }=3\times 2$

$\Rightarrow h^{\prime }=6\text{mm}$


$\text{Reflection at convex mirror}\left(M_2\right):$
The image formed by concave mirror acts as an object for convex mirror. So,

object distance, $u_2=(+60-50)=+10\text{cm}$

Focal length, $f_2=+20\text{cm}$

Using mirror formula:

$\frac{1}{u_2}=\frac{1}{f_2}-\frac{1}{v_2}$

$\Rightarrow \frac{1}{u_2}=\frac{1}{20}-\frac{1}{10}$

$\Rightarrow \frac{1}{u_2}=\frac{-1}{20}$

$\therefore v_2=-20\text{cm}$

Hence, the final position of image after two successive reflection at concave mirror and then at convex $20\text{cm}$ in front of convex mirror, $M_2$.

Magnification:

$m_2=\frac{-v_2}{u_2}=\frac{-(-20)}{10}=2$

So, the image of the point will be formed above $=m_2\times (-6-2)$

$=2\times (-8)\text{mm}$

$=-16\text{mm}$ principal axis of $m_2$


So, the coordinate point of the final image will be
$\left(\right(50+v_2)\text{cm},(-16+2\left)\text{mm}\right)$

$=\left(\right(50-20)\text{cm},-14\text{mm})$

$=(30\text{cm},-14\text{mm})$

Hence, option (a) is the correct answer.

Why this question: To understand the image formation by combination of spherical mirrors.