QJEEM25Ma74e4: JEE Main 2025 Mathematics - Class 11 JEEM25Ma74e4 Solution: Let be the parabola and be its focus. Let be a focal chord of…

$\mathrm{y}^2=12 \mathrm{x} \quad \mathrm{a}=3 \quad \mathrm{SP} \times \mathrm{SQ}=\frac{147}{4}$

Let $\mathrm{P}\left(3 \mathrm{t}^2, 6 \mathrm{t}\right)$ and $\mathrm{t}_1 \mathrm{t}_2=-1$

(ends of focal chord)

So, $Q\left(\frac{3}{t^2}, \frac{-6}{t}\right)$

$S(3,0)$

$\mathrm{SP} \times \mathrm{SQ}=\mathrm{PM}_1 \times \mathrm{QM}_2$

$\begin{aligned}& \text { (dist. from directrix) } \\& =\left(3+3 \mathrm{t}^2\right)\left(3+\frac{3}{\mathrm{t}^2}\right)=\frac{147}{4} \\& \Rightarrow \frac{\left(1+\mathrm{t}^2\right)^2}{\mathrm{t}^2}=\frac{49}{12} \\& \mathrm{t}^2=\frac{3}{4}, \frac{4}{3} \\& \mathrm{t}= \pm \frac{\sqrt{3}}{2}, \pm \frac{2}{\sqrt{3}} \\& \text { considering } \mathrm{t}=\frac{-\sqrt{3}}{2} \\& \mathrm{P}\left(\frac{9}{4},-3 \sqrt{3}\right) \text { and } \mathrm{Q}(4,4 \sqrt{3})\end{aligned}$

$\begin{aligned}&\text { Hence, diametric circle: }\\&\begin{aligned}& (x-4)\left(x-\frac{9}{4}\right)+(y+3 \sqrt{3})(y-4 \sqrt{3})=0 \\& \Rightarrow x^2+y^2-\frac{25}{4} x-\sqrt{3} y-27=0 \\& \Rightarrow \alpha=400, \beta=1728 \\& \beta-\alpha=1328\end{aligned}\end{aligned}$