Question

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

Answer 1

Given: distance between two point sources of light, $d=24cm$, convergent lens focal length, $f=9cm$

To find the position at which the lens should be placed so that the image obtained of both sources is at same point

Solution:

Let the distance between first object and lens be $x$

The the distance between second object and lens will be $24-x$

And the sign convention of image of second object will be negative.

Then, according to lens formula

for first object

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}⟹\frac{1}{9}=\frac{1}{v}+\frac{1}{x}....\left(i\right)$

And for second object,

$\frac{1}{9}=\frac{1}{-v}+\frac{1}{24-x}....\left(ii\right)$

Adding eqn (i) and (ii), we get

$\frac{1}{9}+\frac{1}{9}=\frac{1}{v}+\frac{1}{x}+\frac{1}{24-x}-\frac{1}{v}⟹\frac{2}{9}=\frac{1}{x}+\frac{1}{24-x}⟹\frac{2}{9}=\frac{24-x+x}{x(24-x)}⟹2x(24-x)=9\left(24\right)⟹48x-2x^2=216⟹2x^2-48x+216=0$

divide throughout by 2, we get

$x^2-24x+108=0⟹x(x-18)-6(x-18)=0⟹(x-6)(x-18)=0⟹x=6,x=18$

Hence the lens can be placed at 6cm or 18 cm from the first source in order to get both sources image at same distance.