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
\(m_{\text{tangent}}m_{\text{radius}}=-1\)
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 | |
| Vertical tangents | |
| The \(x\)-axis is a tangent | |
| The \(y\)-axis is a tangent |
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 | |
| Touches the circle; the line is a tangent | |
| Does not meet the circle |
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