Official solution
Family of circles passing through the points (0, 2) and (0, -2)
\[{x^2} + (y - 2)(y + 2) + \lambda x = 0,\,\lambda \in R\]
\[{x^2} + {y^2} + \lambda x - 4 = 0\] ...... (1)
Differentiate w.r.t x
\[2x + 2y{{dy} \over {dx}} + \lambda = 0\] ....... (2)
Using (1) and (2), eliminate \[\lambda\]
\[{x^2} + {y^2} - \left( {2x + 2y{{dy} \over {dx}}} \right)x - 4 = 0\]
\[2xy{{dy} \over {dx}} + {x^2} - {y^2} + 4 = 0\]