How to Use the Circle-Line Intersection Calculator
A circle and a line can only have one of three relationships: they don't meet at all (0 points), they touch at exactly one point (tangent), or they cross at two points. This calculator takes the circle's center (h, k) and radius (r), plus the line's equation ax+by+c=0, and returns both the relationship and the actual intersection coordinates. The circle doesn't need to be centered at the origin — any center works.
The classification is based on comparing the shortest distance from the circle's center to the line, d = |a×h+b×k+c| ÷ √(a²+b²), against the radius r. If d is greater than r, the line misses the circle entirely. If d equals r exactly, the line is tangent, touching at one point. If d is less than r, the line crosses the circle at two points. The actual coordinates come from substituting the line's equation into the circle's equation and solving the resulting quadratic.
This same logic shows up in game development — determining whether a projectile grazes or passes through a circular target — and in road design, where engineers calculate exactly where a curved segment meets a straight one. The radius must be greater than 0, and a and b can't both be 0, since that wouldn't describe a valid line.
Frequently Asked Questions
It means the distance from the circle's center to the line exactly equals the radius, so the line touches the circle at exactly one point.
The line is the equation ax+by+c=0, while the center (h, k) and radius r define the circle (x-h)² + (y-k)² = r².