• # question_answer Two point light sources are 24 cm apart. Where should a convex lens of focal length 9 cm be put in between them from one source so that the images of both the sources are formed at the same place A) 6 cm B) 9 cm C) 12 cm D) 15 cm

[a] The given condition will be satisfied only if one source $({{S}_{1}})$ placed on one side such that $u<f$ (i.e., it lies under the focus). The other source (5y 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 lens, then $\frac{1}{f}=\frac{1}{-y}-\frac{1}{-x}\,\,\Rightarrow \,\,\frac{1}{y}=\frac{1}{x}-\frac{1}{f}$ ....(i)
 If ${{S}_{2}}$ is the object for lens, then $\frac{1}{f}=\frac{1}{+y}\,-\frac{1}{-(24-x)}$ $\Rightarrow$            $\frac{1}{y}=\frac{1}{y}-\frac{1}{(24-x)}$ ...(ii)
 From (i) and (ii), $\frac{1}{x}-\frac{1}{f}=\frac{1}{f}-\frac{1}{(24-x)}$ $\Rightarrow$   $\frac{1}{x}+\frac{1}{(24-x)}=\frac{2}{f}=\frac{2}{9}$ $\Rightarrow$   ${{x}^{2}}-24x+108=0$ On solving the equation, $x=18\text{ }cm,\text{ }6\text{ }cm$