On the assumption that alpha, beta, a and b are real,
$$\alpha -\imath \beta=\frac{1}{a-\imath b}=\frac{a+\imath b}{a^{2}+b^{2}}.$$
Taking the complex conjugate of both sides,
$$\alpha + \imath\beta=\frac{a-\imath b}{a^{2}+b^{2}},$$
and multiplying the equations together,
$$\alpha^{2}+\beta^{2}=\frac{a^{2}+b^{2}}{(a^{2}+b^{2})^{2}},$$
from which the result follows.