Generate a circle centered on a line and touching 2 other circles

We are looking for a circle centered on the given straight line (blue), and touching two given circles (blue). [] If $A$ and $C$ are centers of the given circles, $a$ and $c$ their radii, $K$ the center and $r$ the radius of a touching circle, then $$||KC|-|KA||=|r\pm c-(r\pm a)|=|c-a|.$$ The difference of distances of $K$ to two fixed points is constant. Therefore, the locus of centers of touching circles is a hyperbola with foci $A$ and $C.$ (See also this question.)

One vertex of the hyperbola is $I,$ it lies on $AC$ at equal distance to both blue circles.