the tangent at any point to the circle x^2+y^2+4x-6y-12=0 to the circle x^2+y^2+4x-6y+4=0 is

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$$\begin{gathered} \mathrm{I}\hspace{0.17em}\mathrm{beleive}, \mathrm{you} \mathrm{are} \mathrm{looking} \mathrm{for} \mathrm{common} \mathrm{tangent} \\ \mathrm{For} {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0, \mathrm{centre} \mathrm{is} \left(-\mathrm{g},-\mathrm{f}\right) \mathrm{and} \mathrm{radius} \mathrm{is} \sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}} \\ \mathrm{We} \mathrm{have} \\ {\mathrm{x}}^2+{\mathrm{y}}^2+4\mathrm{x}-6\mathrm{y}-12=0 \\ \mathrm{Centre} \mathrm{is} \left({\mathrm{h}}_1,{\mathrm{k}}_1\right)=\left(-2,3\right) \\ \mathrm{Radius} \mathrm{is} {\mathrm{r}}_1=\sqrt{{\left(-2\right)}^2+3^2+12}=5 \\ {\mathrm{x}}^2+{\mathrm{y}}^2+4\mathrm{x}-6\mathrm{y}+4=0 \\ \mathrm{Centre} \mathrm{is} \left({\mathrm{h}}_2,{\mathrm{k}}_2\right)=\left(-2,3\right) \\ \mathrm{Radius} \mathrm{is} {\mathrm{r}}_2=\sqrt{{\left(-2\right)}^2+3^2-4}=3 \\ \mathrm{As} \mathrm{centre} \mathrm{of} \mathrm{both} \mathrm{circles} \mathrm{are} \mathrm{same}, \mathrm{hence} \mathrm{they} \mathrm{will} \mathrm{be} \mathrm{concentric} \mathrm{which} \\ \mathrm{means} \mathrm{there} \mathrm{will} \mathrm{be} \mathrm{no} \mathrm{common} \mathrm{tangent}. \end{gathered}$$

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