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 the point of concurrecy of the chord of contact AB of the circle `x^(2)+y^(2)=8` is

Options

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

Answer

A

Explanation

(1)
The equation of the circumcenter of `DeltaAQB` is
`x^(2)+y^(2)-4+lamda(xalpha+ybeta-8)=0`
Because it passes through (0,0), i.e., the center of the circle,
`lamda=-(1)/(2)`
Let the circumcenter be (h,k). Then,
`h=(alpha)/(4),k=(beta)/(4)`
`oralpha=4h,beta=4k`
Also, `betat=alpha+2t^(2)`
`oralpha-2beta+8=0" "(becauset=2)`
Substituting `alpha=4handbeta=4k`, we get
h-2k+2=0
Therefore, the locus is x-2y+2=0.