Two circular ponds have the edges x² + y² = 9 and x² + y² - 24x - 10y + 165 = 0, with lengths in…

Step 1. Centres and radii. The first pond has centre $(0, 0)$ and radius $3$ metres. For the second, $g = -12$, $f = -5$ and $c = 165$, so the centre is $(12, 5)$ and the radius is $\sqrt{144 + 25 - 165} = 2$ metres.

Step 2. The distance. $d = \sqrt{144 + 25} = 13$ metres. The sum of the radii is $5$ metres, and $13 > 5$. The ponds are apart, so there are $4$ common tangents.

Step 3. The outer pipe. $\sqrt{d^2 - (r_1 - r_2)^2} = \sqrt{169 - 1} = \sqrt{168} \approx 12.96$ metres.

Step 4. The crossing pipe. $\sqrt{d^2 - (r_1 + r_2)^2} = \sqrt{169 - 25} = 12$ metres.