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.
Answer & explanation
Correct answer: option 1
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