Question

If $z$ be a non-real complex number satisfying $|z|=2$, then which of the following is/are true?

• A. $\arg \left(\frac{z-2}{z+2}\right)=\pm \frac{\pi }{2}$
• B. $\arg \left(\frac{z+1+i\sqrt{3}}{z-1+i\sqrt{3}}\right)=\frac{\pi }{6}$
• C. $\left|z^2-1\right|\ge 3$
• D. $\left|z^2-1\right|\le 5$

Answer 1

Answer: (A), (C), (D)

(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)
$△AOB$ 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 $|z|=2\Rightarrow x,y\in [-2,2]$
${\left|z^2-1\right|}^2={\left|\left(z^2-1\right)\right|}^2=(z+1)|\overline{z}-1|=(\overline{z}+(z+\overline{z})+1)(z\overline{z}-(z+\overline{z})+1)$
$=(5+2x)(5-2x)=25-4x^2$
${\left|z^2-1\right|}^2=25-4x^2\Rightarrow 3\le {\left|z^2-1\right|}^2\le 5$
( $\mathrm{maximum}$ when $x=0$ and minimum when $x=2$ or -2 )