Mathematics

A circle passing through the origin with specific axis intercepts has a special structure: its center must lie on the perpendicular bisectors of the chords it cuts on each axis. Since the circle passes through $(0,0)$ and intercepts length $a$ on the $x$-axis and length $b$ on the $y$-axis, the center is equidistant from the origin and the other endpoints of those intercepts. Use the general circle equation and substitute the three known points (the origin and the two intercept endpoints) to find the center and radius.

A circle passing through the origin with specific axis intercepts has a special structure: its center must lie on the perpendicular bisectors of the chords it cuts on each axis. Since the circle passes through $(0,0)$ and intercepts length $a$ on the $x$-axis and length $b$ on the $y$-axis, the center is equidistant from the origin and the other endpoints of those intercepts. Use the general circle equation and substitute the three known points (the origin and the two intercept endpoints) to find the center and radius.