- #1
WubbaLubba Dubdub
- 24
- 1
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?