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 is 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 upper signs in the two coordinate formulas are taken together, as are the lower signs.
The discriminant
 |
(8)
|
therefore determines the incidence of the line and circle, as summarized in the following table.
 | incidence |
 | no intersection |
 | tangent |
 | intersection |
