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(D) The path difference between the two waves at point $P$ is given by $\Delta x = S_1P - S_2P$.For constructive interference (maxima),the path differ

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$\frac{\sqrt{5}D}{2}$

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(D) The path difference between the two waves at point $P$ is given by $\Delta x = S_1P - S_2P$.For constructive interference (maxima),the path difference must be an integer multiple of $\lambda$,i.e.,$\Delta x = n\lambda$.Given the geometry,the path difference is $\Delta x = S_1S_2 \cos 	heta$,where $S_1S_2 = 3\lambda$.Thus,$3\lambda \cos 	heta = n\lambda$,which implies $\cos 	heta = \frac{n}{3}$.For the smallest distance $x$ (excluding the central maximum at $x=0$),we look for the largest possible value of $n$ such that $n Then $\cos 	heta = \frac{2}{3}$.From the geometry,$\cos 	heta = \frac{D}{\sqrt{x^2 + D^2}}$.Equating the two expressions: $\frac{D}{\sqrt{x^2 + D^2}} = \frac{2}{3}$.Squaring both sides: $\frac{D^2}{x^2 + D^2} = \frac{4}{9}$.$9D^2 = 4x^2 + 4D^2$.$5D^2 = 4x^2$.$x^2 = \frac{5D^2}{4}$.$x = \frac{\sqrt{5}D}{2}$.

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