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Question

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 A 8
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+β216<0
and β24α<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+β216<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).

721597_678889_ans_bf108c9361d440829153c14d1aecab0b.png

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