The delta method and the distance method

There are two routes to establishing the position of a line with respect to a circle. Algebraically, the number of intersection points is obtained by solving the line-circle system, which leads to a second-degree equation: the sign of its discriminant $\Delta$ decides the position. Geometrically, on the other hand, one can use the distance of the centre from the line.

Property — Delta method vs distance method

Let $r$ be a line and $\gamma$ a circle with centre $C$ and radius $R$. Then, equivalently:

\Delta > 0 & \iff & d(C, r) < R \qquad \text{(secante)} \\ \Delta = 0 & \iff & d(C, r) = R \qquad \text{(tangente)} \\ \Delta < 0 & \iff & d(C, r) > R \qquad \text{(esterna)} \end{array}$$ The second method is often quicker because it avoids explicitly solving the system.

It is worth keeping both methods in mind: the delta method is general and also returns the intersection points, whereas the distance method is faster when one only needs to classify the position or impose tangency.

Links

Topics: Analytical circle
Concepts: Discriminant · Point-line distance · Line-circle intersection · Radius · Line · Tangent line
Skills: Analytical geometry · Reasoning by cases