Tangents, Chords and Line Intersections

The perpendicular bisector of a chord passes through the centre C. The tangent at P is perpendicular to the radius CP; the normal at P passes through C. Tangent at a point m_tangentm_radius=-1 Find the radius gradient, take its negative reciprocal and use the point of contact in the line equation. Handle a horizontal or vertical radius directly. Centre (a,b), radius r **Tangent lines** | **Equation or condition** Horizontal tangents | y=b+r and y=b-r Vertical tangents | x=a+r and x=a-r The x-axis is a tangent | Shortest distance from the centre to the x-axis equals the radius: |b|=r The y-axis is a tangent | |a|=r Line–circle intersections Substitute the line into the circle equation and form a quadratic. Its discriminant gives the number of intersections. **The line** | **Discriminant of the quadratic** Cuts the circle at two distinct points | b2-4ac>0 Touches the circle; the line is a tangent | b2-4ac=0 Does not meet the circle | b2-4ac<0