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.>

Distance of the object from the mirror, $u$ = $-$20 cm

Distance of the image in front of the mirror, $v$ = $-$15 cm

To find: Focal length of the mirror $f$.

Solution:

From the mirror formula, we know that-

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

Substituting the given values we get-

$\frac {1}{f}=\frac {1}{(-15)}+\frac {1}{(-20)}$

$\frac {1}{f}=-\frac {1}{15}-\frac {1}{20}$

$\frac {1}{f}=\frac {-4-3}{60}$

$\frac {1}{f}=\frac {-7}{60}$

$f=-\frac {60}{7}cm$

Thus, the focal length of the mirror is $\frac {60}{7}cm$, and the negative sign implies that it is in front of the mirror (on the left). Hence, the mirror is concave.

(b) Given:

Distance of the object from the mirror, $u$ = $-$20 cm

Distance of the image behind the mirror, $v$ = $+$15 cm

To find: Focal length of the mirror $f$.

Solution: From the mirror formula, we know that-

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

Substituting the given values we get-

$\frac {1}{f}=\frac {1}{15}+\frac {1}{(-20)}$

$\frac {1}{f}=\frac {1}{15}-\frac {1}{20}$

$\frac {1}{f}=\frac {4-3}{60}$

$\frac {1}{f}=\frac {1}{60}$

$f=60cm$

Thus, the focal length of the mirror is $60cm$, and the positive sign implies that it is behind the mirror (on the right). Hence, the mirror is convex.