A variable parabola `y^(2) = 4ax, a` (where `a ne -(1)/(4))` being the parameter, meets the curve `y^(2) +x - 2 = 0` at two points. The locus of the point of intersecion of tangents at these points is

Options

A. `x - 2y - 4 = 0`
B. `x - 4y +2 = 0`
C. `x - 4y -1 = 0`
D. `2x - y +1 = 0`

Answer

B

Explanation

The point of intersection `(at^(2),2at)` lies on the curve `y^(2) +x -y -2 =0`.
`:. 4a^(2)t^(2) + at^(2) - 2at -2 =0`
`rarr (4a^(2)+a) t^(2) -2at -2 =0`
Sum of roots, `t_(1) + t_(2) = (2a)/(4a^(2)+a)`
Product of roots, `t_(1)t_(2) =- (2)/(4a^(2)+a)`
Point of intersection of tangents, `(X,y) = (at_(1)t_(2),a(t_(1)+t_(2)))`
`:. x =- (2a)/(4a^(2)+a), y = (2a^(2))/(4a^(2) +a)`
`:. (y)/(x) =- a`
`:. x =- (2(-(y)/(x)))/(4((y)/(x))^(2)+(-(y)/(x)))`
`rArr x = (2xy)/(4y^(2) -xy)`
`rArr 1 = (2)/(4y-x) rArr 4y -x =2`.