Geometry - locus of points

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☕ Hello

I was doing an example when this question came to my mind.

The question I was dealing with: two dots (A and B) exist on the paper. Draw a circle with a radius of X which passes through the points (A and B )

The answer: so I know that the center of the circle must be on the perpendicular bisector of line segment A and B

My question : can the center of the circle be that far which part of the circle becomes exactly on the line And can the center of our circle be that far which it doesn't even pass through the points ?