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Question

If z is a complex number that satisfies z2+z|z|+|z|2=0, then the locus of z is

A
a circle
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B
a straight line
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C
a pair of straight lines
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D
an ellipse
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Solution

The correct option is C a pair of straight lines
z2+z|z|+|z|2=0(z|z|)2+z|z|+1=0
Assuming z|z|=t
t2+t+1=0t=ω,ω2z|z|=ω,ω2z=ω|z| or z=ω2|z|x+iy=|z|(12±i32)x=|z|2, y=±3|z|2
Squaring both the sides,
4x2=x2+y2, 4y2=3(x2+y2)3x2y2=0

This represents a pair of straight lines.

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