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Q.
If $z$ be a non-real complex number satisfying $|z|=2$, then which of the following is/are true?
Complex Numbers and Quadratic Equations
Solution:
(A)
For $P(z), \arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{2}$
and for $Q(z), \arg \left(\frac{z-2}{z+2}\right)=\frac{-\pi}{2}$
$\Rightarrow (A)$ is true.
(B)
$\triangle AOB$ is equilateral.
$\therefore \angle AOB =\frac{\pi}{3} \text { and } \angle ACB =\frac{\pi}{6} $
$\therefore \arg \left(\frac{ z -1+ i \sqrt{3}}{ z +1+ i \sqrt{3}}\right)=\frac{\pi}{6} \text { (By rotation) } $
$\Rightarrow \arg \left(\frac{ z +1+ i \sqrt{3}}{ z -1+ i \sqrt{3}}\right)=-\frac{\pi}{6}$
$\Rightarrow \text { (B) is not true. }$
(C) and (D)
Given $|z|=2 \Rightarrow x, y \in[-2,2]$
$\left| z ^2-1\right|^2=\left|\left( z ^2-1\right)\right|^2=( z +1)|\overline{ z }-1|=(\overline{ z }+( z +\overline{ z })+1)( z \overline{ z }-( z +\overline{ z })+1)$
$=(5+2 x)(5-2 x)=25-4 x^2$
$\left|z^2-1\right|^2=25-4 x^2 \Rightarrow 3 \leq\left|z^2-1\right|^2 \leq 5$
( $\operatorname{maximum}$ when $x =0$ and minimum when $x =2$ or -2 )
