Coordinate Geometry

Line–circle intersections, chords and tangency

• Solve for all paired line–circle intersections and filter segment or ray restrictions.
• Use a repeated root or radius geometry to justify tangency.
• Interpret discriminants, chord lengths and line–quadratic-curve intersections.

Substitute the line into the curve. A quadratic with positive, zero or negative discriminant gives two distinct, one repeated or no real intersection coordinates. Recover the matching other coordinate. A segment or ray can contain fewer points than its supporting line. A perpendicular from the centre bisects a chord.