The number of such points (a+1, √3a) where a is any integer lying inside the region bounded by the circles x²+y²-2x-3=0 and x²+y²-2x-15=0, is

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None of these

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Solution

The correct option is D None of these
The given circles are (x-1)²+y² =4 and (x-1)²+y²=16
The point (a+1, √3a) lie on the line x=a+1, y=√3a

i.e. y=√3(x-1) [eliminating a]

whose slope=√3, hence makes angle 60° with the +ve direction of the X-axis.

Hence, we have

A≡ (1+2cos60°, 2sin60°)≡(2, √3)

B≡(1+4cos60°, 4 sin60°)≡(3, 2√3)

Hence, there is no point on the line segment AB whose abscissa is an integer

Since absicssa of A is 2 and that of B is 3.

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The number of such points (a+1, √3a) where a is any integer lying inside the region bounded by the circles x2+y2-2x-3=0 and x2+y2-2x-15=0, is

The number of such points (a+1, √3a) where a is any integer lying inside the region bounded by the circles x²+y²-2x-3=0 and x²+y²-2x-15=0, is