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

$y=mx+c\Rightarrow \frac{y-mx}{c}=1$

Homogenising $3x^2-y^2-2x+4y=0$ using $\frac{y-mx}{c}=1$

$3x^2-y^2-2x\left(\frac{y-mx}{c} \right)+4y\left(\frac{y-mx}{c} \right)=0$

$\therefore$ since the angel subtended is a right angled 

$\therefore 3c+2m-c+4=0$ $\Rightarrow c=-m-2$

$y=mx+c\Rightarrow y=mx-m-2$

$\Rightarrow y+2=mx-1$

Passing through fixed point (1, -2)