A circle drawn with centre $(α, β),$ then which of the following is/are INCORRECT.

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If $α$ and $β$ are both irrational, then there exist at most one rational point on the circle $x^{2} + y^{2} - 2 α x - 2 β y + c = 0$

If $(α, β) \equiv (e, π),$ then there exist no rational point on the circle $x^{2} + y^{2} - 2 α x - 2 β y + c = 0 \in R$

If $α$ is rational and $β$ is irrational and there exist exactly two rational points P and Q on the circle, then slope of PQ must be 0.

If there exists only one rational point $(x_{1}, y_{1})$ on circle with $(2, \sqrt{3})$ as centre, then $x_{1} = 1$

answer is A, B, D.

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Detailed Solution

If $α, β \in Q^{C} :$
Like irrational then maximum two rational points on the circle
Unlike irrational then maximum number of rational points is one
If $α \in Q a n d β \in Q^{C} \Rightarrow$ Maximum two rational points on the circle located in such a way that they are mirror image of each other about $x = α$ line. For only one rational point, the point should be one end of vertical diameter of the circle
$\Rightarrow x_{1} = 2$