The line and circle intersection explorer takes a line Ax + By = C and a circle (x − h)² + (y − k)² = r². It substitutes one into the other, solves the quadratic that results and tells you whether the line misses, touches or cuts the circle.
It shows the working in the same order you would in an answer, so you can compare it with your own.
How do you use it?
- Enter the line as A, B and C. A horizontal line y = 0 is A = 0, B = 1, C = 0. A vertical line x = 2 is A = 1, B = 0, C = 2.
- Enter the circle’s centre as h and k, and its radius r.
- Press Find intersections.
- Press Reset to return to the sample line y = 0 and circle x² + y² = 9.
If you start with y = mx + c, move the x term across first. The line y = x + 1 becomes −x + y = 1, so A = −1, B = 1, C = 1.
How do you read the result?
The first lines show the quadratic formed by the substitution, in x if B is not 0 and in y if the line is vertical. Next comes the discriminant and its meaning:
- above zero: two points, so the line is a secant;
- zero: one point, so the line is a tangent;
- below zero: no intersection.
A geometric check compares the distance from the centre to the line with the radius. A distance smaller than the radius means two points. Equal to the radius means tangent. Larger means a miss.
If there are intersection points, the tool lists their coordinates and a residual table, which should read about 0 for both the line and the circle. The diagram is drawn to scale with the points marked.
Example walk-through
Take the line y = x + 1, which is A = −1, B = 1, C = 1, and the circle x² + y² = 25, so h = 0, k = 0, r = 5.
The substitution gives 2x² + 2x − 24 = 0, which simplifies to x² + x − 12 = 0. The discriminant of the tool’s version is 4 + 192 = 196, which is positive, so the line cuts the circle twice.
Solving gives x = 3 or x = −4. The y values that go with them are 4 and −3, so the points are (3, 4) and (−4, −3). Check (3, 4): 3² + 4² = 25, and 4 = 3 + 1. Both residuals are 0.
The distance from the centre to the line is 1/√2, about 0.7071, which is less than 5, as expected for two intersections.
Now try y = 3 with the sample circle (A = 0, B = 1, C = 3). The quadratic is x² = 0, the discriminant is zero and the tool reports a tangent at (0, 3). The distance to the line is 3, equal to the radius. Change C to 4 and the discriminant becomes negative, so there is no intersection.
What are the assumptions and limits?
- A and B cannot both be 0. That is not a line.
- The radius must be above 0. A zero or negative radius is not a circle.
- A discriminant within a tiny tolerance of zero is treated as zero, and the tolerance is stated on the page. Residuals below 1e-12 are shown as 0.
- Answers are shown as decimals rounded to four places, not in surd form. In an exam, give exact values if the question asks.
- The tool uses your inputs as given. It does not check that you copied the question correctly.
Which lessons explain the ideas behind it?
- Substitute a line into a circle relationship sets up the quadratic that the tool shows first.
- Find intersections of a line and circle follows the same solving steps in full.
- Use a tangent perpendicular to the radius explains the geometric check.
- Solve a parameter condition for tangency turns the zero-discriminant case into a question about an unknown.
- Circle coordinate methods practice gives mixed questions.
The topic sits in circle coordinate geometry. For a teacher to check your method on real questions, see online one-to-one Additional Mathematics tuition. More tools are in the learning tools directory.