JEE Main 2024 — Parabola Question with Solution

$P(x,y)\&x\ge 3$
Slope of line at $P(x,y)$ will be $\frac{dy}{dx}=\frac{1}{2}\left(\frac{y+5}{x-3}\right)$ $\begin{aligned} & \Rightarrow 2 \frac{d y}{(y+5)}=\frac{1}{(x-3)} d x \\ & \Rightarrow 2 \ln (y+5)=\ln (x-3)+C \end{aligned}$<br/>Passesthrough$(4,-2)$ $\begin{aligned} & \Rightarrow 2 \ln (3)=\ln (1)+C \\ & \Rightarrow C=2 \ln (3) \\ & \Rightarrow 2 \ln (y+5)=\ln (x-3)+2 \ln (3) \\ & \Rightarrow 2\left(\ln \left(\frac{y+5}{3}\right)\right)=\ln (x-3) \\ & \Rightarrow\left(\frac{y+5}{3}\right)^2=(x-3) \end{aligned}$ $\Rightarrow (y+5{)}^2=9(x-3)$
Parabola $\begin{gathered} 4 a=9 \\ a=\frac{9}{4} \end{gathered}$<br/><imgsrc="https://firebasestorage.googleapis.com/v0/b/prepsharp-c91fd.firebasestorage.app/o/je{e}_{m}ains$\begin{aligned} & d=\sqrt{\left(\frac{7}{4}\right)^2+6^2} \\ & d=\frac{\sqrt{625}}{4} \\ & d=\frac{25}{4} \\ & 12 d=75\end{aligned}$