Let a conic C pass through the point (4,-2) and P(x, y), x ≥ 3 , be...

$P(x, y) \& x \geq 3$
Slope of line at $P(x, y)$ will be $\frac{d y}{d x}=\frac{1}{2}\left(\frac{y+5}{x-3}\right)$$\begin{aligned} & \Rightarrow 2 \frac{d y}{(y+5)}=\frac{1}{(x-3)} d x \\ & \Rightarrow 2 \ln (y+5)=\ln (x-3)+C \end{aligned}$<br />Passes through $(4,-2)\begin{aligned} & \Rightarrow 2 \ln (3)=\ln (1)+C \\ & \Rightarrow C=2 \ln (3) \\ & \Rightarrow 2 \ln (y+5)=\ln (x-3)+2 \ln (3) \\ & \Rightarrow 2\left(\ln \left(\frac{y+5}{3}\right)\right)=\ln (x-3) \\ & \Rightarrow\left(\frac{y+5}{3}\right)^2=(x-3) \end{aligned}\Rightarrow(y+5)^2=9(x-3)$<br />Parabola $\begin{gathered} 4 a=9 \\ a=\frac{9}{4} \end{gathered}$<br /><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/9_FUufYi5OdJD2shrSf9IGblPs6QBkRduUJObbXkFz4.original.fullsize.png"><br />\begin{aligned} & d=\sqrt{\left(\frac{7}{4}\right)^2+6^2} \\ & d=\frac{\sqrt{625}}{4} \\ & d=\frac{25}{4} \\ & 12 d=75\end{aligned}$$