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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
a circle of radius 1 unit
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C
a parabola with axis of symmetry parallel to x-axis
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D
a hyperbola
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Solution

## The correct option is B a circle of radius 1 unitImaginary part of z−1+1z−1 is zero. So, z−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)−1=0 As z is not purely real, so (z−1)(¯¯¯z−1)=1 Assuming z=x+iy, then (x−1+iy)(x−1−iy)=1⇒(x−1)2+y2=1 The locus is circle whose centre is (1,0) and radius is 1.

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