JEE Main 2024 — Mathematics Coordinate Geometry

$$P(x, y) \& x \geq 3$$P(x,y)&x≥3

Slope of line at $P(x, y)$P(x,y) will be $\frac{d y}{d x}=\frac{1}{2}\left(\frac{y+5}{x-3}\right)$dxdy​=21​(x−3y+5​) $\begin{aligned}
& \Rightarrow 2 \frac{d y}{(y+5)}=\frac{1}{(x-3)} d x \
& \Rightarrow 2 \ln (y+5)=\ln (x-3)+C
\end{aligned}$

Passes through $(4,-2)$(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 $\begin{gathered}
4 a=9 \
a=\frac{9}{4}
\end{gathered}$

$$\begin{aligned} & d=\sqrt{\left(\frac{7}{4}\right)^2+6^2} \\ & d=\frac{\sqrt{625}}{4} \\ & d=\frac{25}{4} \\ & 12 d=75\end{aligned}$$​d=(47​)2+62​d=4625​​d=425​12d=75​