Question

The distance between two points sources of light is $24cm$. Where should a converging 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

Let,

the lens is placed at $x$ distance from the first source.

for 1st source:

Object distance $u=x$ cm

focal length, $f=9cm$

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$ (from lens formula)

$\Rightarrow \frac{1}{9}=\frac{1}{v}+\frac{1}{x}$

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

for second source:

Object distance, $u=24-x$

focal length, $f=9cm$

$\Rightarrow \frac{1}{9}=\frac{1}{-v}+\frac{1}{24-x}$ (from lens formula)

$\Rightarrow v=9(2y-x)/x-\mathrm{15....}\left(2\right)$

Equating (1) and (2) as LHS is equal,

$\frac{9x}{x-9}=\frac{9(2y-x)}{x-15}$

On solving

$x=6cm$ or $x=18cm$