Find the equation of the circle whose center is $\left(-p, -q\right)$ and radius is $\sqrt{p^2 + q^2}$.
Center $= \left(h, k\right) = \left(-p, -q\right)$
Radius $= r = \sqrt{p^2 + q^2}$
Equation of circle is $\;$ $\left(x - h\right)^2 + \left(y - k\right)^2 = r^2$
i.e. $\;$ $\left(x + p\right)^2 + \left(y + q\right)^2 = \left(\sqrt{p^2 + q^2}\right)^2$
i.e. $\;$ $x^2 + 2px + p^2 + y^2 + 2qy + q^2 = p^2 + q^2$
i.e. $\;$ $x^2 + y^2 + 2px + 2qy = 0$