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Answer

D

Explanation

Plan Parametic coordinates for `y^(2) = 4 ax` are `(at^(2) 2at)`

Descro[topm pf Situation As the circle intersects the parabola at P and Q. Thus, points P and Q should satisfy circle.
`P (2 t^(2), 4t)` should lie on the `x^(2) + y^(2) - 2x - 4y = 0`
`implies 4 t^(4) + 16 t^(2) - 4 t^(2) - 16 t = 0`
`implies 4 t^(4) + 12 t^(2) - 16 t = 0`
`implies 4 t(t^(3) + 2 t - 4) = 0`
`implies 4t ( t - 1) (t^(2) + t + 4) = 0`
`:. t = 0, 1`
`implies P (2,4)` and PQ is the diameter of cirlce
Thus, area of `Delta PQS = (1)/(2). OS xx PQ = (1)/(2) (2).(4) = (4)`