Question

If $z_{1}$ and $z_{2}$ are the two complex roots of equal magnitude and their arguments differ by $\frac{\pi}{2}$, of the quadratic equation $a x^{2}+b x+c=0(a \neq 0)$ then $a$ (in terms of $b$ and $c$ ) is

• A. $\frac{b^{2}}{2 c}$
• B. $\frac{b^{2}}{c}$
• C. $\frac{b}{2 c}$
• D. None of these

Answer 1

Answer: (A)

$z_{1}+z_{2}=-\frac{b}{a} \,\,\, (1)$
$z_{1} z_{2}=\frac{c}{a} \,\,\, (2)$
$z_{2}=i z_{1}\,\,\, (3)$
From Eq. (1) and (2)
$z_{1}(1+i)=\frac{-b}{a}$
$\Rightarrow z_{1}=\frac{-b}{2 a}(1-i)$
$ \Rightarrow z_{1}^{2}=\frac{b^{2}}{4 a^{2}}(-2 i)=\frac{-b^{2}}{2 a^{2}} i$
From Eq. (2) and (3)
$z_{1}^{2}=\frac{c}{a i}=\frac{-c}{a} i $
$\Rightarrow \frac{-b^{2}}{2 a^{2}} i=\frac{c}{a} i $
$\Rightarrow a=\frac{b^{2}}{2 c}$