An (infinite) line determined by two points and may intersect a circle of radius and center (0, 0) in two imaginary points (left figure), a
degenerate single point (corresponding to the line being tangent to the circle; middle
figure), or two real points (right figure).
In geometry, a line meeting a circle in exactly one point is known as a tangent line, while a line meeting
a circle in exactly two points in known as a secant line
(Rhoad et al. 1984, p. 429).
Defining
(1)
(2)
(3)
(4)
gives the points of intersection as
(5)
(6)
where the function is defined as
(7)
The discriminant
(8)
therefore determines the incidence of the line and circle, as summarized in the following table.