An object is placed in front of a concave lens at a distance of 45 cm from it. If its image is formed at a distance of 30 cm from the lens, calculate the focal length of the lens.

Given: concave lens, object distance u = −45 cm, image distance v = −30 cm

(A concave lens always forms a virtual image on the same side as the object, so v is negative.)

Lens formula: 1/f = 1/v − 1/u

1/f = 1/(−30) − 1/(−45) 1/f = −1/30 + 1/45

Taking a common denominator of 90: 1/f = −3/90 + 2/90 1/f = −1/90 f = −90 cm

Focal length of the concave lens = −90 cm (that is, 90 cm, and the minus sign confirms the lens is diverging).

The signs decide this question. The object is on the incoming side, so u = −45. The image from a concave lens is always virtual and on the SAME side as the object, so v = −30, not +30. Putting v in as positive gives f = +18 cm, which would make it a converging lens — impossible for a concave one.

Take the common denominator carefully: −1/30 is −3/90 and 1/45 is 2/90, leaving −1/90.

Then invert. And keep the minus sign in the final answer: a negative focal length is what identifies a diverging lens, and it is part of the answer rather than an accident of the arithmetic.