JEE Main 2024 — 31 January, Shift 2 — Question 21

$\mathrm{P} \equiv\left(\frac{\alpha-2}{2}, \frac{\beta-1}{2}\right) \equiv\left(\frac{\gamma+1}{2}, \frac{\delta}{2}\right)$

$\frac{\alpha-2}{2}=\frac{\gamma+1}{2}$ and $\frac{\beta-1}{2}=\frac{\delta}{2}$ $\Rightarrow \alpha-\gamma=3 \ldots .(1), \quad \beta-\delta=1\ldots .(2)$

Also, $(\gamma, \delta)$ lies on 3x-2y=6 $$$\begin{gathered}3 \gamma-2 \delta=6 \end{gathered}$$and(\alpha, \beta)$lies on$2 x-y=5$

$\Rightarrow 2 \alpha-\beta=5 \ldots .(4)$

Solving (1), (2), (3), (4) $\alpha=-3, \beta=-11, \gamma=-6, \delta=-12$

$|\alpha+\beta+\gamma+\delta|=32$