Two ways to decide
A line can meet a circle in two points, one point or none.
The distance method is faster when you only need the position. The system gives the actual points.
The tangent line at a point
If you know the point of tangency \(P\), use the fact that a tangent is perpendicular to the radius at \(P\).
Chord length
The perpendicular from the center bisects a chord. That makes a right triangle with hypotenuse \(r\) and one leg \(d\), so
\[\text{chord length} = 2\sqrt{r^2 - d^2}\]
Practice
Quick check
The distance from the center of a circle to a line is less than the radius. How many points do they share?
When \(d < r\), the line is a secant.
Quick check
The tangent to a circle at a point is always:
A tangent meets the radius at the point of tangency at a right angle.