JEE Main 2024 — 8 April, Shift 2 — Question 27

$a|x|=|y| e^{y x-\beta}, a, b \in N$

$x d y-y d x+x y(x d y+y d x)=0$

$\frac{d y}{y}-\frac{d x}{x}+(x d y+y d x)=0$

$\ell \operatorname{nn}|\mathrm{y}|-\ln |\mathrm{x}|+\mathrm{xy}=\mathrm{c}$

$\mathrm{y}(1)=2$

$\ell \mathrm{n}|2|-0+2=\mathrm{c}$

$\mathrm{c}=2+\operatorname{\ell n} 2$

$\ell \mathrm{n}|\mathrm{y}|-\ell \mathrm{n}|\mathrm{x}|+\mathrm{xy}=2+\ell \mathrm{n} 2$

$\ln |x|=\ln \left|\frac{y}{2}\right|-2+x y$

$|x|=\left|\frac{y}{2}\right| e^{x y-2}$

$2|\mathrm{x}|=|\mathrm{y}| \mathrm{e}^{\mathrm{xy}-2}$

$\alpha=2, \quad \beta=2, \quad \alpha+\beta=4$