All the chords of the hyperbola $3 x^2-y^2-2 x+4 y=0$, subtending a right angle at the origin pass through the fixed point

Let $a x+b y=1$ be the chord ....(i) Making the equation of hyperbola homogeneous using Eq. (i), we get $3 x^2-y^2+(-2 x+4 y)(a x+b y)=0$ or $\quad(3-2 a) x^2+(-1+4 b) y^2+(-2 b+4 a) x y=0$ Since, the angle subtended at the origin is a right angle. $\therefore$ Coefficient of $x^2+$ Coefficient of $y^2=0$ $\begin{array}{ll}\Rightarrow & (3-2 a)+(-1+4 b) \\ \Rightarrow & =0 \\ \Rightarrow & a=2 b+1\end{array}$ $\therefore$ Chords are $(2 b+1) x+b y-1=0$ or $\quad b(2+y)+(x-1)=0$, which clearly pass through the fixed point $(1,-2)$.