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Question

Determine the locus of the point z such that z2z1 is always real.

A
x(x2+y22x)=0.
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B
y(x2+y2+2x)=0.
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C
y(x2+y22x)=0.
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D
x(x2+y2+2x)=0.
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Solution

The correct option is D y(x2+y22x)=0.
We know that if z is real, then z=¯z If z2z1 is real then z2z1=¯z2¯z1
Cross multiplying, we get (z¯z){z¯z(z+¯z})=0

z¯z=0 or z=¯z z is real y=0 or z¯z(z+¯z)=0 or x2+y22x=0

Hence the locus is either y=0 i.e x-axis or a circle x2+y22x=0

Note : You may do it by Cartesian method and put imaginary part equal to zero.
y(x2+y22x)=0.
Ans: C

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