A small circular lake x² + y² - 2x - 4y - 4 = 0 lies inside a larger lake x² + y² - 8x - 12y - 12 =…

Step 1. The circles. The small lake has centre $(1, 2)$ and radius $\sqrt{1 + 4 + 4} = 3$ kilometres. The large lake has centre $(4, 6)$ and radius $\sqrt{16 + 36 + 12} = 8$ kilometres.

Step 2. They touch. $d = \sqrt{9 + 16} = 5$ kilometres and $8 - 3 = 5$. The circles touch from inside.

Step 3. The contact point. It divides $C_1C_2$ externally in the ratio $3 : 8$. So $\left(\dfrac{8 \times 1 - 3 \times 4}{8 - 3},\ \dfrac{8 \times 2 - 3 \times 6}{8 - 3}\right) = \left(-\dfrac{4}{5}, -\dfrac{2}{5}\right) = (-0.8, -0.4)$. Check: its distance from $(1, 2)$ is $\sqrt{3.24 + 5.76} = 3$.

Step 4. The common tangent. $S_1 - S_2 = 6x + 8y + 8 = 0$, which is $3x + 4y + 4 = 0$. It passes through $(-0.8, -0.4)$, since $-2.4 - 1.6 + 4 = 0$. At $x = 0$, $y = -1$ kilometre.