$$C_1+\lambda \,C_2=0$$
are all circles passing through intersection of
$$C_1=0,C_2=0$$
Is it possible to express orthogonal trajectories of all the $C_1+\lambda \,C_2=0 $ circles using another parameter $\mu?$
Attempts with complex algebra and bipolar coordinates have not been lucky.
EDIT 1:
I took circles in the form
$$ \dfrac{x^2+y^2 + 2 f_1 x +2g_1 y +c_1}{x^2+y^2 + 2 f_2 x +2g_2 y +c_2} = -\lambda $$ by differentiating and simplifying got it into form
$$ \dfrac{dy}{dx}=\dfrac{y+k}{x+h}$$
and integrated to get straight lines
$$ (y+k) = \mu\, (x+h)$$
but find this wrong.