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

Question

Let a conic $C$ pass through the point $(4,-2)$ and $P(x,y),x\ge 3$, 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 $12d$ equals \_\_\_\_\_\_\_\_.

Answer: 75

Accepted Answer

As per given condition

$\begin{gathered} \frac{dy}{dx}=\frac{y+5}{2(x-3)} \ \Rightarrow \ln (y+5)=\frac{1}{2}\ln (x-3)+c \ (4,-2)\Rightarrow \ln 3=\frac{1}{2}\ln 1+c \ \Rightarrow c=\ln 3 \end{gathered}$

$\Rightarrow$ Curve is $(y+5{)}^{2}=9(x-3)$

Focal distance of $(7,1)=\frac{9}{4}+4=\frac{25}{4}=d$

$12d=75$

Solutions · 1