Visible and Invisible Intersections in the Cartesian Plane

The circle and the straight line are plotted on the Cartesian plane. Their two intersections and , shown as black points, represent solutions of the simultaneous equations. But suppose the value of is such that the two curves do not intersect? There are still two perfectly good algebraic solutions of the simultaneous equations: , but these now involve complex numbers. You can imagine that the two intersections are now invisible.


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