Question

Let $p, q, r \,\in\, R$ and $r > p > 0$. If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta$, then $\left|\alpha\right|+\left|\beta\right|$ is

• A. equal to 1
• B. less than 2 but not equal to 1
• C. greater than 2
• D. equal to 2

Answer 1

Answer: (C)

Given quadratic equation is
$px^{2}+qcx+r=0\,...\left(1\right)$
$D=q^{2}-4pr$
Since $\alpha$ and $\beta$ are two complex root
$\therefore \beta=\bar{\alpha} \Rightarrow \left|\beta\right|=\left|\bar{\alpha}\right| \Rightarrow \left|\beta\right|=\left|\alpha\right|$
$\left(\because \left|\bar{\alpha}\right|=\left|\alpha\right|\right)$
Consider
$\left|\alpha\right|+\left|\beta\right|=\left|\alpha\right|+\left|\alpha\right|\, \left(\because\left|\beta\right|=\left|\alpha\right|\right)$
$=2\left|\alpha\right| > 2.1=2\,\left(\because \left|\bar{\alpha }\right|> 1\right)$
Hence, $\left|\alpha \right|+\left|\beta \right|$ is greater than $2.$