Circle and Line Intersection Calculator
Find whether a circle and an infinite line meet in zero, one or two points. Compare the center-to-line distance, projection foot and intersection coordinates with a diagram.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Compute S=Ah+Bk+C and N=A²+B². Compare r²N with S² exactly to classify intersections. Project the center onto the line; for two intersections, move from that foot in both directions along the line.
Circle: (x−h)²+(y−k)²=r²; line: Ax+By+C=0; D=r²(A²+B²)−(Ah+Bk+C)²
Worked example
For x²+y²=25 and y−3=0, the line is 3 units from the center and the radius is 5. There are two intersections, (−4,3) and (4,3). Changing the line to y−5=0 gives the tangent point (0,5); y−6=0 gives no intersection.
Assumptions and input details
This solves an infinite straight line, not a bounded segment. Inputs are whole numbers, fractions or decimals with up to six places, within magnitude 1e6 and reduced denominator 1e6. Radius must be positive; A and B cannot both be zero. The sign of D is computed exactly, so near-tangent inputs are not automatically rounded into tangency. Coordinates, distances and the equal-scale diagram use numerical approximations. The diagram shows a square neighborhood extending 1.4 radii from the center; a sufficiently distant line is outside that view and is identified as such.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
The number of intersections and projection-foot fractions are exact for the accepted inputs. Square-root coordinates, distances and diagram positions are approximations.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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