A circular lake has the equation (x - 2)² + (y - 1)² = 10, with lengths in kilometres…

Step 1. The distance condition. The line is $x - y + c = 0$. Its distance from $(2, 1)$ is $\dfrac{|2 - 1 + c|}{\sqrt{2}} = \dfrac{|1 + c|}{\sqrt{2}}$. This must equal $\sqrt{10}$.

Step 2. Solve. $|1 + c| = \sqrt{20} = 2\sqrt{5}$, so $c = -1 \pm 2\sqrt{5}$. That is $c \approx 3.47$ or $c \approx -5.47$.

Step 3. The distance between the roads. Two parallel tangents lie on opposite sides of the centre, so they are a diameter apart: $2\sqrt{10} \approx 6.32$ kilometres. Check with the formula for parallel lines: $\dfrac{|3.47 - (-5.47)|}{\sqrt{2}} = \dfrac{8.94}{1.414} \approx 6.32$.