Question

The distance between two point sources of light is 24cm .Where should a convergent lens witha focal length of f=9cm be placed between them to obtain the images of both sources at thesame point?

Answer 1

Answer:

Thus the lens should be placed either at 6 cm or at 18 cm between the two light sources.

Explanation:

Given

The distance between the two light sources = 24 cm

Focal length of the convergent lens f = 9 cm

Let the lens is at u distance from the first light source

using

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

$\frac{1}{9} =\frac{1}{u}+\frac{1}{v}$

$\implies \frac{1}{v} =\frac{1}{9}-\frac{1}{u}$

Similarly for the second light source

$\frac{1}{9} =\frac{1}{24-u}-\frac{1}{v}$

$\implies \frac{1}{9}=\frac{1}{24-u}-\frac{1}{9}+\frac{1}{u}$

$\implies \frac{2}{9}-\frac{1}{u}=\frac{1}{24-u}$

$\implies \frac{2u-9}{9u}=\frac{1}{24-u}$

$\implies (2u-9)(24-u)=9u$

$\implies 48u-216-2u^2+9u=9u$

$\implies 2u^2+216-48u=0$

$\implies u^2-24u+108=0$

$\implies u^2-18u-6u+108=0$

$\implies u(u-18)-6(u-18)=0$

$\implies (u-18)(u-6)=0$

$\implies u=6, 18$

Thus the lens should be placed either at 6 cm or at 18 cm between them.