Two circular lakes have edges x² + y² - 4x - 6y - 12 = 0 and x² + y² - 16x - 6y + 48 = 0, with…

Step 1. The common chord. Subtracting, $12x - 60 = 0$, so $x = 5$. So $k = 5$.

Step 2. The centres and radii. The first circle has centre $(2, 3)$ and radius $\sqrt{4 + 9 + 12} = 5$ kilometres. The second has centre $(8, 3)$ and radius $\sqrt{64 + 9 - 48} = 5$ kilometres. The distance between the centres is $6$ kilometres.

Step 3. The length. The distance of $(2, 3)$ from $x = 5$ is $3$ kilometres. The chord is $2\sqrt{25 - 9} = 8$ kilometres.