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Offline Centres

Consider the following curves:
$C_{1} : y^{2} = 4 x$
$C_{2} : y^{2} (a + b x) = {(a + c x)}^{2}, a, b, c, \in I^{+}, (\forall x, y w h e r e e v e r d e f i n e d)$
Where $C_{2}$ is the locus of the centre of family of circles which passes through focus of $C_{1}$ and cut $C_{1}$ orthogonally.
On the basis of above information, answer the following questions:

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