1.3.4—Lines and circles

Substitute the line equation into the circle equation. The resulting quadratic has two roots for a secant, one repeated root for a tangent, and no real roots when the line misses the circle.

The repeated-root condition is equivalent to the perpendicular distance from the centre to the line equalling the radius. Use whichever route makes the parameter condition clearer.

Substituting y=mx+c into a circle gives Δ=0 at tangency; the repeated x-value then gives the contact point.

A discriminant classifies real algebraic intersections, but a point must also satisfy any stated segment or domain restriction.