JEE Main 2024 — 27 January, Shift 2 — Question 4

$P\left(a^{2}, a+1\right)$ $\mathrm{L}_{1}=3 \mathrm{x}-\mathrm{y}+1=0$

Origin and P lies same side w.r.t. $\mathrm{L}_{1}$ $\Rightarrow \mathrm{L}_{1}(0) . \mathrm{L}_{1}(\mathrm{P})>0$

$\therefore 3\left(a^{2}\right)-(a+1)+1>0$

$\Rightarrow 3 \mathrm{a}^{2}-\mathrm{a}>0$

$\mathrm{a} \in(-\infty, 0) \cup\left(\frac{1}{3}, \infty\right)$

Let $L_{2}: x+2 y-5=0$

Origin and P lies same side w.r.t. $\mathrm{L}_{2}$

$\Rightarrow \mathrm{L}_{2}(0) \cdot \mathrm{L}_{2}(\mathrm{P})>0$

$\Rightarrow \mathrm{a}^{2}+2(\mathrm{a}+1)-5<0$

$\Rightarrow \mathrm{a}^{2}+2 \mathrm{a}-3<0$

$\Rightarrow(\mathrm{a}+3)(\mathrm{a}-1)<0$

$\therefore \mathrm{a} \in(-3,1)$

Intersection of (1) and (2) $\mathrm{a} \in(-3,0) \cup\left(\frac{1}{3}, 1\right)$