Two point light sources are 24cm apart. At what distance from one source, a convex lens of focal length 9cm be kept in between them, so that the images of both the sources are formed at the same place.

6 cm

The given condition will be satisfied only if one source (S₁) placed on one side such that u < f (i.e. it lies under the focus). The other source (S₂) is placed on the other side of the lens such that u > f (i.e. it lies beyond the focus).

If S₁ is the object for lens then

$\frac{1}{f}=\frac{1}{-y}-\frac{1}{-x} \Rightarrow \frac{1}{y}=\frac{1}{x}-\frac{1}{f}$   .................(i)

If S₂ is the object for lens then

$\frac{1}{f}=\frac{1}{+y}-\frac{1}{-(24-x)} \Rightarrow \frac{1}{y}=\frac{1}{f}-\frac{1}{(24-x)}$    ..........(ii)

From equation (i) and (ii)

$\frac{1}{x}-\frac{1}{f}=\frac{1}{f}-\frac{1}{(24-x)}$

$\frac{1}{x}+\frac{1}{(24-x)}=\frac{2}{f}=\frac{2}{9}$

$\Rightarrow x^2-24 x+108=0$

On solving we have x = 18 cm, 6 cm