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)
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(2)
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(3)
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(4)
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gives the points of intersection as
(5)
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(6)
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where the function is defined as
(7)
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The discriminant
(8)
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therefore determines the incidence of the line and circle, as summarized in the following table.
incidence | |
no intersection | |
tangent | |
intersection |