- #1

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## Homework Statement

$$z^2 + z|z| + |z|^2=0$$

The locus of ##z## represents-

a) Circle

b) Ellipse

c) Pair of Straight Lines

d) None of these

## Homework Equations

##z\bar{z} = |z|^2##

## The Attempt at a Solution

Let ##z = r(cosx + isinx)##

Using this in the given equation

##r^2(cos2x + isin2x) + r^2(cosx + isinx) + r^2 = 0##

##r^2(cos2x + cosx + 1 + i(sin2x + sinx)) =0##

Thus ##r=0## or ##cos2x + cosx + 1 = 0## and ##sin2x + sinx =0##

I can't think of a geometrical interpretation of this result. Using the real part as x and the imaginary part as y makes a funny graph at wolfram alpha

But solving analytically, I get a pair of straight lines

##z^2 + z\sqrt{z\bar{z}} + z\bar{z}=0##

##z(z + \sqrt{z\bar{z}} + \bar{z})=0##

Thus, either## z = 0## or ##(z + \sqrt{z\bar{z}} + \bar{z})=0##

In case ##z + \sqrt{z\bar{z}} + \bar{z} =0##

##2Re(z) = -\sqrt{z\bar{z}}##

Let ##z = x + iy##

##(\sqrt{3}x - y)(\sqrt{3}x + y)=0##

Where did I go wrong?