If a circle has points $A$ and $B$ which lie on the coordinate axis and $AB$ is the diameter of the circle, would it always form a perfect circle where the midpoint of $AB$ is the centre of the circle?
The question states the line $y = -3x + 12$ meets the coordinate axis at $A$ and $B$. It asks you to find the equation of a circle that passes through points $A, B$and $O$ where $O$ is the origin.
The midpoint of $AB$ is $(2,6)$ so would this be the centre of the circle and if so, is it the same for all lines that meet the coordinate axis?
Since $\angle AOB=\dfrac{\pi}{2}$, the circle must be centered at the midpoint of $AB$, which, as you say, would be the diameter of the circle. And the angle stays unchanged for random line intersecting both coordinate axis.