Circle

The general equation of a circle is: Here, represents the center of the circle, is its radius, and is any point on the circle.Given that the circle's center lies on the line , we have .We can substitute this relationship and use the following two points through which the circle passes: and .By setting the distances equal, we obtain: Simplifying, we have: This simplifies to: Solving for , we find: The circle's center is with radius (the same as the distance from to ).Next, we verify which given points lie on this circle:Observing the point , it results in an irrational distance, hence it can be dismissed. Similarly, results in an irrational value, ruling it out.Checking the remaining options: Thus, the point lies on the circle.

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To determine the locus of the centers of circles passing through the origin with a fixed radius, let's analyze the general equation of a circle: Here, represents the center of the circle, and is the radius.Given that the circle passes through the origin , we substitute and into the circle's equation: This simplifies to: The equation describes a circle with a radius of and centered at the origin . Hence, the locus of the centers of such circles is a circle itself, centered at the origin with the same fixed radius .