Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$,…

Question

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$, then $12 d$ equals ______

Aicharya · Accepted Answer

$P(x, y) \& x \geq 3$
Slope of line at $P(x, y)$ will be $\frac{d y}{d x}=\frac{1}{2}\left(\frac{y+5}{x-3}\right)$ $\begin{aligned} & \Rightarrow 2 \frac{d y}{(y+5)}=\frac{1}{(x-3)} d x \\ & \Rightarrow 2 \ln (y+5)=\ln (x-3)+C \end{aligned}$
Passes through $(4,-2)$ $\begin{aligned} & \Rightarrow 2 \ln (3)=\ln (1)+C \\ & \Rightarrow C=2 \ln (3) \\ & \Rightarrow 2 \ln (y+5)=\ln (x-3)+2 \ln (3) \\ & \Rightarrow 2\left(\ln \left(\frac{y+5}{3}\right)\right)=\ln (x-3) \\ & \Rightarrow\left(\frac{y+5}{3}\right)^2=(x-3) \end{aligned}$ $\Rightarrow(y+5)^2=9(x-3)$
Parabola $\begin{gathered} 4 a=9 \\ a=\frac{9}{4} \end{gathered}$

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

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope…

Solution