Find the differential equation of all circles in the plane orthogonal to a fixed circle/disk boundary of radius $R$
$$ (x-h)^2 + (y-k)^2 =R^2 \tag1 $$
and show how they are related to the Poincaré disk model expressed as:
$$ {\dfrac{ds^2}{T^2}}= \dfrac{dx^2+dy^2}{T^2-x^2-y^2} \tag2 $$
where
$$ T^2= h^2+k^2+R^2 \tag3 $$
is the power of the boundary circle.
( As is well known, if in (1) $R$ is taken varying arbitrarily, by differentiating and setting $y^{'}\rightarrow \frac{-1}{y^{'}}$ and integrating we obtain all radial lines orthogonal to it with an arbitrary slope $ \frac{y-k}{x-h} = m ). $
EDIT1/2:
Twice differentiating 1) w.r.t. $x$ results in $ y^{''}= \dfrac{1+y^{'2}}{k-y}, $ but looks not entirely right, the same is re-derived to include constant $R$ as:
$$ y^{''}= \dfrac{y^{'}(1+y^{'2})}{x-h}= \pm \dfrac{y^{'}(1+y^{'2})}{\sqrt{R^2-(y-k)^2}} \, ; $$
Still unable to include given constants $(h,k)$ in the final DE equation, so please help.