A bridge deck runs from A(-2, 3) to B(6, -3), with lengths in metres…

Step 1. The equation. $(x + 2)(x - 6) + (y - 3)(y + 3) = 0$. This is $x^2 - 4x - 12 + y^2 - 9 = 0$, so $x^2 + y^2 - 4x - 21 = 0$.

Step 2. The radius. The centre is $(2, 0)$. The radius is $\sqrt{4 + 0 + 21} = 5$ metres. Check: $AB = \sqrt{64 + 36} = 10$ metres, and half of it is $5$.

Step 3. The lamp post. At $P(5, 4)$: $25 + 16 - 20 - 21 = 0$. So $P$ is on the circle.

Step 4. The angle. $\overrightarrow{PA} = (-7, -1)$ and $\overrightarrow{PB} = (1, -7)$. The dot product is $-7 + 7 = 0$, so the angle $APB$ is $90^\circ$.