A circular tank has the equation x² + y² + 6x - 2y + 6 = 0, with lengths in metres…

Step 1. The circle. $g = 3$, $f = -1$ and $c = 6$. The centre is $(-3, 1)$ and $r = \sqrt{9 + 1 - 6} = 2$ metres.

Step 2. The pipe length. $S_1 = 4 + 1 + 12 - 2 + 6 = 21$. So each pipe is $\sqrt{21} \approx 4.58$ metres.

Step 3. The chord of contact. $T = 0$ is $2x + y + 3(x + 2) - (y + 1) + 6 = 0$, which is $5x + 11 = 0$, so $x = -2.2$. Its distance from the centre $(-3, 1)$ is $0.8$ metres. The chord is $2\sqrt{4 - 0.64} \approx 3.67$ metres.

Step 4. The area. The distance from $P$ to the chord is $2 - (-2.2) = 4.2$ metres. The area is $\tfrac{1}{2} \times 3.666 \times 4.2 \approx 7.70$ square metres. The formula gives the same: $\dfrac{rL^3}{r^2 + L^2} = \dfrac{2 \times 21\sqrt{21}}{25} \approx 7.70$.