Two coherent point sources  ${\text{S}}_{\text{1}}{\text{ and S}}_{\text{2}}$ vibrating in phase emit light of wavelength  $\lambda$. The separation between the sources is $2\lambda$. The smallest distance from ${\text{S}}_{\text{2}}$ on a line passing through  ${\text{S}}_{\text{2}}$ and perpendicular to  ${\text{S}}_{\text{1}}{\text{,S}}_{\text{2}}$, where a minimum of intensity occurs is:

[Concept]: Path difference at S₂ is $2\lambda$. Therefore for minimum intensity(dark band) at P.

  $\begin{array}{l}For Dark band:path difference \Delta x=\left(\frac{2n-1}{2}\right)\lambda \\ \text{}{S}_{1}P-S_{2}P=\frac{3\lambda }{2}\mathrm{..........}\left(i\right)\end{array}$
  $\sqrt{4{\lambda }^2+x^2}-x=\frac{3\lambda }{2}$

 $\begin{array}{l}\Rightarrow \sqrt{4{\lambda }^2+x^2}=x+\frac{3\lambda }{2}\text{}\\ squaring on bothsides\text{}\\ \Rightarrow {\left(\sqrt{4{\lambda }^2+x^2}\right)}^2={\left(x+\frac{3\lambda }{2}\right)}^2\\ \text{}\Rightarrow 4{\lambda }^2+x^2=x^2+{\left(\frac{3\lambda }{2}\right)}^2+2x\frac{3\lambda }{2}\text{}\\ \Rightarrow 4{\lambda }^2-\frac{9{\lambda }^2}{4}=3x\lambda \text{}\\ \Rightarrow \left(16-9\right)\frac{{\lambda }^2}{4}=3x\lambda \end{array}$     
Solving equation, $x=\frac{7\lambda }{12}$