All about Complex Conjugate

Let $\alpha = p+qi, \beta = r+si$ such that $\alpha + \bar{\beta} = (p+qi)+(r-si) = -7 -9i+7 = -9i$ , or

$p+r=0$ (i),

$q-s=-9$ (ii).

Expanding out $\alpha\bar{\alpha} + \alpha\beta +\bar{\alpha}\bar{\beta} + \beta\bar{\beta}$ now yields:

$\alpha\bar{\alpha} + \alpha\beta +\bar{\alpha}\bar{\beta} + \beta\bar{\beta} = (p+qi)(p-qi)+(r+si)(r-si)+(p+qi)(r+si)+(p-qi)(r-si)$ ;

or $p^2 + q^2 +r^2 + s^2 + 2pr - 2qs + (ps+qr)i - (ps+qr)i;$

or $p^2 + q^2 +r^2 + s^2 + 2pr - 2qs$ ;

or $(p+r)^2 + (q-s)^2$ ;

or $0^2 + (-9)^2 = \boxed{81}.$