Line and circle intersections

Intersection points satisfy both equations. Substitution reduces the problem to one quadratic; distinct roots are distinct points, a repeated root is tangency, and no real roots mean the line misses the circle. Method: substitute the line into the circle, simplify and solve. Recover both coordinates and check them. To classify without solving fully, use the discriminant; handle a vertical line by substituting its fixed x value directly. Example: x2+y2=25 and y=3 give x2=16, hence (-4,3) and (4,3). The joining segment is a chord. Exam check: report coordinate pairs, not just roots. y=5 has one point (0,5); y=6 has none. A full secant line contains a chord segment.