Put . Squaring the two nonnegative moduli and expanding gives . Completing the square yields
(1)
Thus the locus is an Apollonius circle with center and radius . No extraneous points were introduced by squaring, because both original sides are nonnegative. Its intersections with the imaginary axis are and , useful landmarks for the sketch. It surrounds zero but does not pass through zero:
Figure 1.
Apollonius circle centered at three eighths i with radius nine eighths and imaginary-axis intercepts minus three quarters i and three halves i
.

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