A concave mirror has a focal length of 20 cm. At what distance from the mirror should a 4 cm tall object be placed so that it forms an image at a distance of 30 cm from the mirror? Also, calculate the size of the image formed.

(a) Given:

It is a concave mirror.

Focal length of the mirror, $f$ = $-$20 cm

Image distance from the mirror, $v$ = $-$30 cm  

Height of the object, $h$ = $+$4 cm

To find: Object distance from the mirror $u$.

Solution:

From the mirror formula, we know that-

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

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

Substituting the given values in the mirror formula we get-

$\frac{1}{u}=\frac{1}{(-20)}-\frac{1}{(-30)}$

$\frac{1}{u}=-\frac{1}{20}+\frac{1}{30}$

$\frac{1}{u}=\frac{-3+2}{60}$

$\frac{1}{u}=-\frac{1}{60}$

$u=-60cm$

Therefore, the object should be placed at a distance of 60 cm from the mirror.

Now, 

From the magnification formula, we know that-

$m=\frac {-v}{u}=\frac {h'}{h}$

Substituting the required value we get-

$\frac {-(-30)}{-60}=\frac {h'}{4}$

$-\frac {1}{2}=\frac {h'}{4}$

$h'=-\frac {4}{2}$

$h'=-2cm$

Thus, the size of the image formed is 2 cm, and the minus sign implies that it is inverted.