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Question

If z is a complex number not purely real such that imaginary part of z−1+1z−1 is zero, then locus of z is-

A
a straight line parallel to x-axis
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B
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C
a parabola with axis parallel to x-axis
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D
a rectangular hyperbola
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Solution

The correct option is C a circle of radius 1z−1+1z−1=¯¯¯¯¯¯¯¯¯¯¯¯z−1+1¯¯¯¯¯¯¯¯¯¯¯¯z−1⇒ z−¯¯¯z+1z−1−1¯¯¯z−1=0⇒(z−¯¯¯z)−¯¯¯z−1−z+1(z−1)(¯¯¯z−1)=0⇒(z−¯¯¯z)(1−1(z−1)(¯¯¯z−1))=0⇒ z=¯¯¯z or (z−1)(¯¯¯z−1)=1But z is not purely real (z−1)(¯¯¯z−1)=1⇒ z¯¯¯z−z−¯¯¯z=0 which is a circle of unit radius.

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