My problem is that A and B are clearly fixed points, so how can the centre of the circle be a locus of it centre is also fixed? Is there something wrong with the question or am I missing an Important detail?
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No, if $AB$ is a diameter, the center of the circle is always the midpoint of $AB$. If $AB$ is instead a chord (random guess), the locus would be the perpendicular bisector of $AB$. – player3236 Mar 04 '21 at 03:30
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It is hard to come up with a convincingly correct answer to a poorly worded problem. If $A,B$ were not meant to be the endpoints of the diameter, the locus of center would be the line through $A,B$. – cosmo5 Mar 04 '21 at 06:25