Determine the locus of the point z such that z2z−1 is always real.
A
x(x2+y2−2x)=0.
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B
y(x2+y2+2x)=0.
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C
y(x2+y2−2x)=0.
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D
x(x2+y2+2x)=0.
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Solution
The correct option is Dy(x2+y2−2x)=0. We know that if z is real, then z=¯z If z2z−1 is real then z2z−1=¯z2¯z−1 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+y2−2x=0
Hence the locus is either y=0 i.e x-axis or a circle x2+y2−2x=0
Note : You may do it by Cartesian method and put imaginary part equal to zero. ∴y(x2+y2−2x)=0. Ans: C