What this calculator does
Intersection of a circle and a line works out where a line y = mx + c crosses a circle centred at (h, k). Enter your own figures above and the answer updates as you type: nothing is fixed in the code, so the result reflects exactly the numbers you supply.
The formula this calculator evaluates is printed under the tool and explained below, so you can check the working by hand or reuse it in a spreadsheet.
The formula
FormulaSubstitute the line into (x−h)²+(y−k)²=r² to get a quadratic Ax²+Bx+C=0 with A=1+m², B=2(m(c−k)−h), C=h²+(c−k)²−r²
The inputs explained
| Field | What to enter |
|---|---|
| Circle centre h | A number. Starts at 0. |
| Circle centre k | A number. Starts at 0. |
| Radius r | A number. Starts at 5. |
| Line slope m | A number. Starts at 1. |
| Line y-intercept c | A number. Starts at 0. |
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the answer changes with circle centre h
Every other input is held at the calculator’s starting values while circle centre h varies. Select any row to load that scenario into the calculator.
How the answer changes with circle centre h
| Circle centre h | Point 1 | Point 2 | Discriminant |
|---|---|---|---|
| 1 | (4.000, 4.000) | (-3.000, -3.000) | 196.000 |
| 2 | (4.391, 4.391) | (-2.391, -2.391) | 184.000 |
| 5 | (5.000, 5.000) | (0, 0) | 100.000 |
| 10 | No intersection: the line misses the circle | -200.000 | N/A |
| 20 | No intersection: the line misses the circle | -1,400.00 | N/A |
| 50 | No intersection: the line misses the circle | -9,800.00 | N/A |