If the equation of the common tangent at the point $$\left( {1,\, - 1} \right)$$  to the two circles, each of radius $$13,$$  is $$12x + 5y - 7 = 0$$    then the centers of the two circles are :

Solution :
Let $$A,\,B,$$  be the centers of the two circles,
Slope of the common tangent $$ = - \frac{{12}}{5}$$

$$\therefore $$  Slope of $$AB$$  is $$\tan \,\theta = - \frac{1}{{ - \frac{{12}}{5}}} = \frac{5}{{12}}$$
The point $$\left( {1,\, - 1} \right)$$  lies on the line $$AB$$  and the points $$A$$ and $$B$$ are at a distance $$13$$  from the point $$\left( {1,\, - 1} \right)$$
$$\therefore $$  Coordinates of $$A$$ and $$B$$ are $$\left( {1 \pm 13\,\cos \,\theta ,\, - 1 \pm 13\,\sin \,\theta } \right),$$      where $$\tan \,\theta = \frac{5}{{12}}$$
$$\eqalign{ & {\text{i}}{\text{.e}}{\text{.,}}\,\,\left( {1 \pm 13\frac{{12}}{{13}},\, - 1 \pm 13\frac{5}{{13}}} \right){\text{ or }}\left( {1 \pm 12,\, - 1 \pm 5} \right) \cr & {\text{i}}{\text{.e}}{\text{.,}}\,\,\left( {13,\,4} \right){\text{ and }}\left( { - 11,\, - 6} \right) \cr} $$