Let the locus of the centre α , β , β > 0 , of the circle which touches the circle x 2 + y - 1 2 = 1 externally and also touches the x -axis be L . Then the area bounded by L and the line y = 4 is

Step-by-step solution

Since the circle with centre α , β touches the circle x 2 + y - 1 2 = 1 externally and also touches x - axis , therefore the distance between the centres will be equal to the sum of the radius of both circles i.e. α - 0 2 + β - 1 2 = β + 1 ⇒ α 2 = 4 β Hence the locus of L will be x 2 = 4 y Now, the area bounded by x 2 = 4 y & y = 4 will be A = 2 ∫ 0 4 4 - x 2 4 d x = 2 4 x - x 3 12 0 4 = 64 3