Write . Since , the two loci are
(1)
Thus they are radius- circles centred at and , each missing the origin. Their circle equations intersect at and , but only the latter belongs to the original loci. At that point the radius vectors are and , so the tangents meet at a right angle.
Every point on either circle has distance at most from the origin. Equality occurs only at on the first or on the second. The unit circle is therefore internally tangent to both and crosses neither. The sketch marks the excluded origin explicitly, rather than treating it as a second intersection.
Figure 1.
Reciprocal complex loci and orthogonal principal axes of a hyperbola
. Original exact geometric plots. The empty point at the origin is excluded from both reciprocal loci; the hyperbola uses the original level six, with its principal axes and asymptotes. Details.

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