Question

Where should a convex lens of focal length $9cm$ be placed $\left(incm\right)$ between two point sources $S_1$ and $S_2$ which are $24cm$ apart, so that images of both the sources are formed at the same place? Find distance of lens from $S_1$ or $S_2$ whichever is lesser.

Answer 1

Let the distance from the lens to the first source be $x$. Then distance from lens to second source is $24-x$

Applying lens formula for $S_2$,

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

For $S_1$,

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\frac{1}{v}+\frac{1}{x}=\frac{1}{9}$
$\frac{1}{v}=\frac{1}{9}-\frac{1}{x}$

Equating the two,

$\frac{1}{9}-\frac{1}{24-x}=\frac{1}{9}-\frac{1}{x}$
$x^2-24x+108=0$
i.e $x=6$ or $18$

The lesser value is $x=6$.