Jee Main 2024 29 01 2024 Shift 2 - Question7

Answer: (1)

Solution:

Chord with point, $T=S _1$

$\Rightarrow x x _1-4\left(y+y _1\right)=x _1^{2}-8 y _1$

$\left(x _1, y _1\right) \equiv\left(1, \frac{5}{4}\right) \Rightarrow x-4\left(y+\frac{5}{4}\right)=\frac{1-8 \times 5}{4}$

$x-4 y-5=-9$

$\Rightarrow x-4 y+4=0$

$(\alpha, \beta)$ lie on $(L 1)$ and also $y^{2}=4 x$

$\Rightarrow \alpha-4 \beta+4=0$

$$ \begin{aligned} & \beta^{2}=4 \alpha \\ & \beta^{2}=4(4 \beta-4) \\ & \beta^{2}-16 \beta+16=0 \end{aligned} $$

$$ \begin{aligned} & \Rightarrow(\beta-8)^{2}=64-16=48 \\ & \Rightarrow \beta=8 \pm 4 \sqrt{3} \\ & \alpha=4 \beta-4 \\ &= 28 \pm 16 \sqrt{3} \\ &(28+16 \sqrt{3}, 8+4 \sqrt{3}) \text { and }(28-16 \sqrt{3}, 8-4 \sqrt{3}) \\ &(8-\beta)(\alpha-28) \\ & \Rightarrow(-4 \sqrt{3})(16 \sqrt{3}) \\ &=-192 \end{aligned} $$