Question

An object placed 20 cm in front of a mirror is found to have an image 15 cm (a) in front of it, (b) behind the mirror. Find the focal length of the mirror and the kind of mirror in each case.

Answer 1

(i) Distance of the object 'u' = -20 cm

Image is formed in front of the mirror at a distance (v) of -15 cm.

Therefore, the focal length of the mirror 'f' is

$$\begin{gathered} \Rightarrow \frac{1}{f}=\frac{1}{v}+\frac{1}{u} \\ \Rightarrow \frac{\mathit{1}}{\mathit{f}}=\frac{1}{-15}+\frac{1}{-20} \\ \Rightarrow \frac{\mathit{1}}{\mathit{f}}=-\frac{4}{60}-\frac{3}{60}=\frac{-7}{60} \\ \Rightarrow f=\frac{-60}{7}\mathrm{cm} \end{gathered}$$

Thus, the mirror is concave and has a focal length of 60/7.

(ii) Distance of the object 'u' = -20 cm

Image is formed behind the mirror at a distance (v) of 15 cm.

Then, the focal length of the mirror 'f' will be

$$\begin{gathered} \frac{\mathit{1}}{f}\mathit{=}\frac{\mathit{1}}{v}\mathit{+}\frac{\mathit{1}}{u} \\ \Rightarrow \frac{1}{f}=\frac{1}{15}+\frac{1}{-20} \\ \Rightarrow \frac{1}{f}=\frac{4}{60}-\frac{3}{60}=\frac{1}{60} \\ \Rightarrow f=60\mathrm{cm} \end{gathered}$$

Thus, the mirror is convex and has a focal length of 60 cm.