The number of points with integral coordinates that lie in the interior of the region common to the circle x2+y2=16 and the parabola y2=4x, is
A
8
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B
10
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C
16
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D
None of these
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Solution
The correct option is A8 Let (α,β) be the point with integral coordinates and lying in the interior of the region common to the circle x2+y2=16 and the parabola y2=4x. Then, α2+β2−16<0 and β2−4α<0 It is clear from the figure that 0<α<4 ⇒α=1,2,3[∵αϵZ] When, α=1 β2<4α ⇒β2<4 ⇒β=0,1 So, the points are (1,0) and (1,1). When α=2 β2<4α ⇒β2<8 ⇒β=0,1,2 So, the points are (2,0),(2,1) and (2,2). When α=3 β2<4α ⇒β2<12 ⇒β=0,1,2,3 So, the points are (3,0),(3,1),(3,2) and (3,3) Out of these points, (3,3) does not satisfy α2+β2−16<0.
Thus, the points lying in the region are (1,0),(1,1),(2,0),(2,1),(2,2),(3,0)(3,1) and (3,2).