Complex Numbers 2 Question 41
42.
Find all non-zero complex numbers $z$ satisfying $\bar{z}=i z^{2}$.
(1996, 2M)
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Answer:
Correct Answer: 42. $(z=i, \pm \frac{\sqrt{3}}{2}-\frac{i}{2})$
Solution:
- Let
$ \begin{aligned} z & =x+i y . \\ \bar{z} & =i z^{2} \\ (\overline{x+i y}) & =i(x+i y)^{2} \\ x-i y & =i\left(x^{2}-y^{2}+2 i x y\right) \\ x-i y & =-2 x y+i\left(x^{2}-y^{2}\right) \end{aligned} $
NOTE: It is a compound equation, therefore we can generate from it more than one primary equations.
On equating real and imaginary parts, we get
$x =-2 x y \text { and }-y=x^{2}-y^{2} $
$\Rightarrow x+2 x y =0 \text { and } x^{2}-y^{2}+y=0 $
$\Rightarrow x(1+2 y) =0 $
$\Rightarrow x =0 \text { or } y=-1 / 2$
When $x=0, x^{2}-y^{2}+y=0 \Rightarrow 0-y^{2}+y=0$
$\Rightarrow y(1-y)=0 $
$\Rightarrow y=0 $ or $ y=1$
When, $y=-1 / 2, x^{2}-y^{2}+y=0$
$\Rightarrow x^{2}-\frac{1}{4}-\frac{1}{2} =0 $
$\Rightarrow x^{2}=\frac{3}{4} $
$\Rightarrow x = \pm \frac{\sqrt{3}}{2}$
Therefore, $z=0+i 0,0+i ; \pm \frac{\sqrt{3}}{2}-\frac{i}{2}$
$\Rightarrow \quad z=i, \pm \frac{\sqrt{3}}{2}-\frac{i}{2}$
