All chords of a curve $3x^2-y^2-2x+4y=0$ which subtends a right angle at origin passing through fixed point 

Given Equation  $3x^2-y^2-2x+4y=0$ --(1)

Let Equation of chord is y = mx + c

$\begin{array}{l}\Rightarrow \frac{y-mx}{c}=1--\left(2\right)\\ Homoginozing\left(1\right)with\left(2\right)\\ 3x^2-y^2-2x\left(1\right)+4y\left(1\right)=0\\ \Rightarrow 3x^2-y^2-2x\left(\frac{y-mx}{c}\right)+4y\left(\frac{y-mx}{c}\right)=0\\ \Rightarrow 3x^2-y^2-\frac{2xy}{c}+\frac{2m{x}^2}{c}+\frac{4y^2}{c}-\frac{4mxy}{c}=0\\ Thelinesare\perp riff\end{array}$

$\begin{array}{l}co.effof{x}^2+co.effof{y}^2=0\\ \Rightarrow 3-1+\frac{2m}{c}+\frac{4}{c}=0\\ \Rightarrow 2c+2m+4=0\\ \Rightarrow m+2+c=0\\ comparingwithmx-y+c=0\\ x=1,y=-2\\ \therefore Thechordspassesthrough\left(1,-2\right)\end{array}$