Circular curves of the 3rd class obtained by pedal transformation in the quasi-hyperbolic plane

In the quasi-hyperbolic plane the pedal transformation with respect to a given polar line p maps a point T into the pedal line which passes through the point T and its perpendicular point incident with the polar line p. The pedal curve cp of a given generating curve c with respect to the polar line p is the locus of the pedal lines of all the points of the curve c. The pedal curves of quasi-hyperbolic conics that result with a circular curve of the 3rd class will be given and the conditions to obtain such a curve will be shown.