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Show that all chords of the curve 3x^2-y...

Show that all chords of the curve `3x^2-y^2-2x+4y=0,` which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

Text Solution

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The given curve is
`3x^(2)-y^(2)-2x+4y=0`…..`(i)`
Let `y=mx+c` be the chord of curve `(i)` which subtend right angled at origin. Then, the combined equation of lines joining points of intersection of curve `(i)` and chord `y=mx+c` to the origin, can be obtained by the equation of the curve homogenous, i.e.
`3x^(2)-y^(2)-2x((y-mx)/(c ))+4y((y-mx)/(c ))=0`
`implies3cx^(2)-cy^(2)-2xy+2mx^(2)+4y^(2)-4mxy=0`
`implies(3c+2m)x^(2)-2(1+2m)y+(4-c)y^(2)=0`
Since, the lines represented are perpendicular to each other.
`:.` Coefficient of `x^(2)+` Coefficient of `y^(2)=0`
`implies3c+2m+4-c=0`
`impliesc+m+2=0`
On comparing with `y=mx+c`
`implies y=mx+c` passes through `(1,-2)`.
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