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Auxiliary Circle of Ellipse

The number of...

Question

The number of lattice points (lattice points are points with both coordinate integers) on the circle with centre (199, 0) and radius 199 is___

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Solution

Equation of circle is $(x - 199)^{2} + y^{2} = 199^{2}$
$x = 199 + 199 c o s θ, y = 199 s i n θ s i n θ = 1 \Rightarrow (199, 199) s i n θ = 0 \Rightarrow (398, 0), (0, 0)$
$s i n θ = - 1 \Rightarrow (199, - 199)$ are the only points.

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