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.

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` subtending 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 "Coeff. of x"^(2)+"Coeff. of y"^(2)=0]`
`rArr" "a-2b=1`
`rArr" "ax+by=1` always passes through `(1,-2)`