Question

If $\alpha ,\beta$ are complex number such that $|\alpha +\beta {|}^2=|\alpha {|}^2+|\beta {|}^2+2k$, then $k=$

• A. $Re\left(\overline{\alpha }\beta \right)$
• B. $Re(\alpha +\beta )$
• C. $Re\left(\overline{\alpha }\overline{\beta }\right)$
• D. $Re(\overline{\alpha }+\overline{\beta })$

Answer 1

Answer: (A)

$Re\left(\overline{\alpha }\beta \right)$

we know that $z.\overline{z}={\left|z\right|}^2$
$\therefore {|\alpha +\beta |}^2=(\alpha +\beta )\times \left(\overline{\alpha +\beta }\right)$
$=(\alpha +\beta )\times (\overline{\alpha }+\overline{\beta })$
$=\alpha \overline{\alpha }+\beta \overline{\beta }+\overline{\alpha }\beta +\overline{\beta }\alpha$
$={\left|\alpha \right|}^2+{\left|\beta \right|}^2+\overline{\alpha }\beta +\overline{\beta }\alpha$
comparing with the given equation
$Re(\overline{\alpha }\beta +\overline{\beta }\alpha )=2k$
$\therefore k=Re\frac{(\overline{\alpha }\beta +\overline{\beta }\alpha )}{2}$

$\Rightarrow k=Re\left(\overline{\alpha }\beta \right)$ (The real part of $\overline{\alpha }\beta and\overline{\beta }\alpha$ are same as they are conjugate of each other)