Suppose a parabola `y = x^(2) - ax-1` intersects the coordinate axes at three points A,B and C, respectively. The circumcircle of `DeltaABC` intersects the y-axis again at the point `D(0,t)`. Then the value of `t` is

Options

A. `1//2`
B. `1`
C. `3//2`
D. `2`

Answer

B

Explanation

We have `x = (a+-sqrt(a^(2)+4))/(2)`
`:. alpha =(a+sqrt(a^(2)+4))/(2), beta =(a-sqrt(a^(2)+4))/(2)`
Equation of family of circles through A and B is
`(x- alpha) (x-beta) + y^(2) + lambda y =0`
As it passes through `C(0,-1)` we have
`alpha beta +1 -lambda =0` (But `alpha beta =-1)`
`rArr -1 + 1- lambda =0`
`rArr lambda =0`
`:.` Equation of circle through A,B and C is
`(x-alpha) (x-beta) +y^(2) =0`
It cuts the y-axis when `x =0`, so
`alpha beta +y^(2) =0`
`y^(2) =1` (Put `alpha beta =-1)`
`rArr y =1` or `-1`
Hence `t =1`