Final answer:
The equation of the circle is (x - a/2)² + (y - b/2)² = (a/2)² + (b/2)², where the circle passes through the origin and the intercepts on the axes are (a, 0) and (0, b).
Step-by-step explanation:
The equation of a circle passing through the origin (0, 0) and making intercepts a and b on the x-axis and y-axis respectively can be determined using the general equation of a circle (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius. Since the circle crosses the axes at (a, 0) and (0, b), we can infer that the center of the circle is at the point (a/2, b/2) because of symmetry, and the radius is the distance from the center to one of these intercepts, calculated using the Pythagorean theorem.
Lets represent the equation as (x - a/2)² + (y - b/2)² = r². The distance from (a/2, b/2) to the origin is r = √((a/2)² + (b/2)²). Therefore, plugging this into the equation, we get (x - a/2)² + (y - b/2)² = (a/2)² + (b/2)², which is the required equation of the circle.