Enter the first center (a, b) and radius r, then the second center (c, d) and radius s. Use abstract numeric coordinates in one common coordinate system. Both radii must be strictly positive, and the centers must be distinct: a ≠ c or b ≠ d. Check these requirements before interpreting the results; this calculator does not automatically classify concentric circles or enforce these combined conditions.
The outputs are the coefficients of Ax + By + C = 0. Keep their signs when substituting them into the equation. The equal-power line exists for distinct centers even when the circles do not intersect.
The powers of (x, y) are (x − a)² + (y − b)² − r² and (x − c)² + (y − d)² − s². Equating them and subtracting the second expression from the first cancels x² and y², leaving Ax + By + C = 0. Distinct centers ensure that A and B are not both zero. The line is perpendicular to the line joining the centers.
For two intersecting circles, it extends their common chord. For tangent circles with distinct centers, it is their common tangent at contact. See University of Washington, Math 445 Assignment 6, Problem 6.2.
| First circle: center; radius | Second circle: center; radius | A | B | C | Radical axis |
|---|---|---|---|---|---|
| (0, 0); 3 | (6, 0); 3 | 12 | 0 | -36 | 12x − 36 = 0, or x = 3 |
| (0, 0); 1 | (6, 0); 1 | 12 | 0 | -36 | x = 3, although the circles do not intersect |
| (1, 2); 3 | (5, 4); 2 | 8 | 4 | -41 | 8x + 4y − 41 = 0 |
Concentric circles are outside this calculator’s required input domain. If a = c and b = d, then A = B = 0 and C = s² − r². Unequal positive radii give no equal-power points; equal positive radii describe the same circle, so every point has equal power. Neither case defines a unique radical-axis line. These are explanations, not automatic classifications by the calculator.
Zero radii are outside the intended domain even though the numeric input bounds allow zero. Enter strictly positive radii.
The outputs are coefficients, not intersection coordinates or chord lengths. Multiplying all three coefficients by the same nonzero number preserves the line. Results display up to eight decimal places, so a very small coefficient may round to zero. Large inputs can exceed the numeric range, and subtracting nearly equal large quantities can lose precision. An unavailable result is not zero.