Question

Roots of the equation $x^{4}-2 x^{2}+4=0$ forms a

• A. rectangle
• B. square
• C. rhombus
• D. paralleleogram

Answer 1

Answer: (A)

$x^{4}-2 x^{2}+4=0$
$\therefore x^{2}=\frac{2 \pm \sqrt{4-16}}{2}$
$=1 \pm \sqrt{3} i=2 \frac{1 \pm \sqrt{3} i}{2}=2\left(\cos \frac{\pi}{3} \pm i \sin \frac{\pi}{3}\right)$
$\therefore x=\pm \sqrt{2}\left(\cos \frac{\pi}{3} \pm i \sin \frac{\pi}{3}\right)^{1 / 2}$
$=\pm \sqrt{2}\left(\cos \frac{\pi}{6} \pm i \sin \frac{\pi}{6}\right)$
$\therefore x=\sqrt{2} e^{i \pi / 6},-\sqrt{2} e^{i \pi / 6}, \sqrt{2} e^{-i \pi / 6},-\sqrt{2} e^{-i \pi / 6}$
Clearly these points forms rectangle inscribed in circle of radius $\sqrt{2}$.