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

B

Explanation

Given equation of the curve is
` 3x^ 2 - y ^ 2 - 2x + 4y = 0 `… (i)
Let, the equation of one of the chord be ` y = mx + c `
` rArr ( y - mx ) / ( c ) = 1 `…(ii)
On making Eq. (i) homogeneous , we get
` 3x ^ 2 - y ^ 2 + ( - 2 x + 4y ) (( y - mx ) /( c)) = 0 `
`rArr x ^ 2 ( 3c + 2m ) + y ^ 2 ( - c+ 4 ) - 2x y - 4mxy = 0 `
Which represents a pair of straight lines passing through origin. since, the angle subtended is a right angle.
` therefore 3c + 2m - c + 4 = 0 `
` rArr c = m - 2 `
On subsituting value of c in ` y = mx + c`, we have
` y = mx - m - 2 `
` rArr y + 2 = m ( x - 1 ) `
` rArr ` All such chords pass through a fixed point (1, - 2 ).