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Tangent Circles

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Question

Find maximum number of points having integer coordinates (both $x, y$ integer) which can lie on a circle with centre at $(\sqrt{2}, \sqrt{3})$ is (are)

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Solution

$(\sqrt{2}, \sqrt{3})$ is a rational coordinate. No $3$ points with integer coordinates have irrational circumcentre.
Hence, maximum integral point $= 2$

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