locus of the centres of the circles touching the line 3x - 4y +1 =0 and 12x + 5y -3 =0

it should be same as the combined equation of the lines bisecting the angle between given pair of lines

the perpendiculars drawn from any point on the angle bisectors of any two lines meets the lines at equal distance.

this distance will the radius of the circle needed.

so the required locus is the pair of angle bisectors of the two lines.

3x + 11y  - 4 = 0