All chords of the curve 3x²-y²-2x+4y=0 which subtend a right angle at the origin, passing through the fixed point    

$\mathrm{y}=\mathrm{mx}+\mathrm{c}\Rightarrow \frac{\mathrm{y}-\mathrm{mx}}{\mathrm{c}}=1$$\to \left(1\right)$

Given pair of strairht line 3x²-y²-2x+4y=0$\to \left(2\right)$

Homogenising (2) with help of (1)

$3x^2-y^2-2x\left(\frac{y-mx}{c}\right)+4y\left(\frac{y-mx}{c}\right)=0$
$\therefore$since the angle subtend is a right angled then
$\therefore$3c+2m-c+4=0$\Rightarrow$c+m+2=0
comparing with  y=mx+c
$\Rightarrow$x=1,y=-2
Passing through the  fixed point (1, -2)