Official solution
Let equation of circle is
x2 + y2 + 2fx + 2fy + e = 0, it passes through (0, 2b)
\[ \Rightarrow \] 0 + 4b2 + 2g \[ \times \] 0 + 4f + c = 0
\[ \Rightarrow \] 4b2 + 4f + c = 0 . . . (i)
\[2\sqrt {{g^2} - c} = 4a\] . . . (ii)
g2 - c = 4a2 \[ \Rightarrow \] c = \[\left( {{g^2} - 4{a^2}} \right)\]
Putting in equation (1)
\[ \Rightarrow \] 4b2 + 4f + g2 - 4a2 = 0
\[ \Rightarrow \] x2 + 4y + 4(b2 - a2) = 0, it represent a hyperbola.