Let $C_1$ be the curve $xy=1$ and $C_2$ be the reflection of $C_1$ in the line $y=2x$. let the equation of $C_2$ can be written in the form of $12x^2+bxy+c{y}^2+d=0$ then which of the following is TRUE?

Let $P_1(u,v)$ be any point of $C_1$ and $P_2(x,y)$ be the corresponding point on $C_2$ 
Since, $P_1{P}_2$ (line joining) is perpendicular to $y=2x$ so,
Its slope  $=-\frac{1}{2}\mathrm{.....}\left(1\right)$
And midpoint of $P_1{P}_2$ lies on $y=2x$

$\frac{y+v}{2}=2\left(\frac{x+u}{2}\right)\mathrm{.....}\left(2\right)$

Solving equation (1) and (2), we get  $12x^2-7xy-12y^2+25=0$