Q.Why did we use λ as the path difference between the waves originating from the extreme points of a single slit for the first minimum?

For a single slit of width a illuminated by a plane wavefront, dividing the slit into pairs of matching sub-sources shows that whenever the path difference between the two extreme edge rays, $a\sin\theta$, equals a whole wavelength $n\lambda$, the ENTIRE slit's light cancels in matching pairs, giving a diffraction MINIMUM at $a\sin\theta=n\lambda$. The (approximate) maxima fall roughly midway between successive minima, near $a\sin\theta=(n+\tfrac12)\lambda$. Using the small-angle approximation ($D\gg a$), this gives fringe spacing $W=\lambda D/a$ -- identical in FORM to Young's double-slit fringe width, but with slit WIDTH a replacing slit SEPARATION d -- with the crucial exception that the central bright fringe is TWICE as wide, $2\lambda D/a$, as every other fringe, since it spans between the first minima on either side.

A real lab double-slit interference pattern (30-40 near-equal fringes) looks very different from a real single-slit diffraction pattern (20-30 fringes, brightest and widest at the centre, tapering outward) -- and a realistic double slit of finite width shows BOTH effects combined: fine interference fringes whose heights are modulated by the broader single-slit diffraction envelope.