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

All chords of the curve `3x^2-y^2-2x+4y=0` which subtend a right angle

1.	All chords of the curve `3x^2-y^2-2x+4y=0` which subtend a right angle at the origin, pass through the fixed point
Answer» The given curve is `3x^(2)-y^(2)-2x+4y=0` (1) Let `y=mx+c` be the chord of combined equations of lines joining the points of intersction of curve (1) and chord `y=mx+c`to the origin can be obtained by making the equation of curve homogeneous with the help of the equation of chord as `3x^(2)-y^(2)-2x((y-mx)/(c))+4((y-mx)/(c))=0` or `3cx^(2)-cy^(2)-2xy+2mx^(2)+4y^(2)-4mxy=0` or `(3c+2m)x^(2)-2(1+2m)xy+(4-c)y^(2)=0` As the lines represented by this pair are perpendicular to each other , we must have Coeffcient of `x^(2)+`Coefficient of `y^(2)=0` (1) Hence , `3c+2m+4-c=0` or `-2=m+c` Comparing this result with `y=mx+c` , we can see that `y=mx+c` (2) passes though (1,-2).

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