PROF65 wrote:

My construction is based on the Gergonne construction of the fourth circle tangent to three given circles:
Let $S$ be the similicenter of $r,s$ ; $T$ touch point of a tangent to $r$ through $S$ ;
the pole of the line through $S$ perpendicular to $\ell$ wrt $r$ is $R_1$ ;
$d$ , the radical axis of $r,s$ , cuts $\ell$ at $R$ ;
$P$ a point of intersection of $RR_1$ and $r$ ;
$TP$ cuts $d$ at $K$ ; let the bisector of
$KP$ cuts $\ell$ at $O$
the circle with center $O$ passing through $K$ is tangent to $r,s$ .
Best regards.
RH HAS

Very nice sir. In my sol I introduced $R, S$ the centers of $r, s$ and their radii be $r_1, s_1$ .The point $O$ on $l$ with satisfy $OR-OS=r_1-s_1$ is the required center of circle. This is done by marking intersection of hyperbola (having foci $R, S$ and constant $r_1-s_1$ ) with line $l$ .It is easy to find synthetic sol for this.