Find the co-ordinates of image of point object P formed after two successive reflections in figure, considering the first reflection at concave mirror and then at convex mirror.

The ray diagram for the situation is drawn in figure (but not to scale).

For reflection at concave mirror $M_1$, we have

$$\begin{gathered} u=-20 \mathrm{cm} \\ {f}_1=-15 \mathrm{cm} \end{gathered}$$

Since,$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$$\begin{gathered} \Rightarrow \frac{1}{v_1}+\frac{1}{(-20)}=\frac{1}{(-15)} \\ \Rightarrow v_1=-60 \mathrm{cm} \end{gathered}$$

So, magnification$m_1=-\frac{v_1}{u}=-\frac{-60}{-20}=-3$ (Inverted)

$\Rightarrow A^{\text{'}}{P}^{\text{'}}=m_1\left(AP\right)=3\times 2=6 \mathrm{mm}$

For reflection at convex mirror $M_2$, we have

$$\begin{gathered} u=+10 \mathrm{cm} \\ {f}_2=+20 \mathrm{cm} \end{gathered}$$

Since $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$$\begin{gathered} \Rightarrow \frac{1}{v_2}+\frac{1}{10}=\frac{1}{20} \\ \Rightarrow v_2=-20 \mathrm{cm} \end{gathered}$$

Again, magnification,$m_2=-\frac{v_2}{u}=-\frac{(-20)}{10}=2$