The locus of the point satisfying , where is a non-zero real number, is a circle with its center on the -axis. The equation represents an arc of a circle passing through and , with the center located at .

Locus Type: Circle.

Center Location: -axis .

Property: Since the points and lie on the real axis ( -axis), and the argument is a constant , the center of the circle must lie on the perpendicular bisector of the segment connecting -1 and 1 , which is the imaginary axis ( axis).

Therefore, the correct option is a circle with center on y-axis.

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