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  4. The equation z2+ barz2-2|z|2+z+ barz=0 represents a

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Q. The equation $z^{2}+\bar{z}^{2}-2|z|^{2}+z+\bar{z}=0$ represents a

Complex Numbers and Quadratic Equations

Solution:

We have, $z^{2}+\bar{z}^{2}-2|z|^{2}+z+\bar{z}=0$
$\Rightarrow (x+i y)^{2}+(x-i y)^{2}-2\left(x^{2}+y^{2}\right)+x+i y+x-i y=0$
$($ Putting $z=x+i y)$
$\Rightarrow 2 x^{2}+2(i y)^{2}-2 x^{2}-2 y^{2}+2 x=0$
$\Rightarrow -4 y^{2}+2 x=0 $
or $y^{2}=\frac{1}{2} x$
which is a parabola.