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Two point sources $S_1$ and $S_2$ are $24 \mathrm{~cm}$ apart. Where should a convex lens of focal length $9 \mathrm{~cm}$ be placed in between them, so that the images of both sources are formed at the same place?
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Verified Answer
The correct answer is:
12 cm
The given condition will be satisfied only if one source $\left(S_1\right)$ placed on one side such that $u < f$ (i.e. it lies under the focus. The other source $\left(S_2\right)$ is placed on the other side of the lens such that $u>f$ (i.e. it lies beyond the focus). If $S_1$ is the object for the lens, then
If $S_2$ is the object for the lens, then
Adding $(i) \&(i i)$, we get
$$
\begin{aligned}
& \frac{1}{x}+\frac{1}{(24-x)}=\frac{2}{f}=\frac{2}{7} \\
& \Rightarrow \quad x^2-24 x+108=0 \Rightarrow x=18 \mathrm{~cm} \text { and } x=6 \mathrm{~cm}
\end{aligned}
$$
So, the correct answer will be $6 \mathrm{~cm}$.
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