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

Options

A. `(1,2)`
B. `(-1,2)`
C. `(1,-2)`
D. `(-1,-2)`

Answer

C

Explanation

Let `ax+by=1` be a family of chords of the curve `3x^2-y^2-2x+4y=0` substending a right angle at the origin.
The combined equation of the straight lines joining the origin to the points of intersection of `3x^2-y^2-2x+4y=0` and the chord `ax+by=1` is `3x^2-y^2-(2x-4y)(ax+by_=0`
This represents a pair of perpendicular lines.
`therefore 3-2a-1+4b=0 " "[because "Coefficient of " x^2+ "Coefficient of" y^2=0]`
`rArr a-2b=1`
So, `ax+by=1` always passes through `(1,-2)`.