A tangent is drawn at any point P(t) on the parabola `y^(2)=8x` and on it is takes a point `Q(alpha,beta)` from which a pair of tangent QA and OB are drawn to the circle `x^(2)+y^(2)=8`. Using this information, answer the following questions :
The locus of circumcenter of `DeltaAQB` id t=2 is

Options

A. `y^(2)-2x=0`
B. `y^(2)-x^(2)=4`
C. `y^(2)+4x=0`
D. `y^(2)-2x^(2)=4`

Answer

C

Explanation

(3)
The equation of the tangent at point P of the parabola `y^(2)=8x` is
`yt=x+2t^(2)` (1)
The equation of the chord of contact of the circle `x^(2)+y^(2)=8` w.r.t. `Q(alpha,beta)` is
`xalpha+ybeta=8` (2)
`Q(alpha.beta)` lies on (1). Hence,
`betat=alpha+2t^(2)` (3)
`:.xalpha+y((alpha)/(t)+2t)-8=0` [From (2) and (3)]
`or2(ty-4)+alpha(x+(y)/(t))=0`
For point of concurrency,
`x=-(y)/(t)andy=(4)/(t)`
Therefore, the locus is `y^(2)+4x=0`