A circle S = 0 touches a circle ${\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+6\mathrm{y}-23=0$ internally and the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+8\mathrm{y}+19=0$ externally. The locus of centre of the circle S = 0 is conic whose eccentricity is k then $\left[\frac{{\displaystyle 1}}{{\displaystyle \mathrm{k}}}\right]$ is where [.] denotes G.I.F