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Intercept of a Line with a Circle

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Date: 05/28/99 at 06:24:11
From: Steven Doyle
Subject: Intercept of a line with a circle

Given the line,

ax + by + c = 0

and a circle of arbitrary center and radius,

x^2 + y^2 - 2fx - 2gy + d = 0

where,
d = f^2 + g^2 - r^2
with,
(f,g): Circle's Center

What is the general equation for the intercept between this circle and
line? I know it should be some sort of quadratic answer (as you can
have 0, 1, or 2 intercept points) but I can't seem to derive it.
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Date: 05/28/99 at 08:05:16
From: Doctor Jerry
Subject: Re: Intercept of a line with a circle

Hi Steven,

A general equation can be found (see below; I did it with
Mathematica), but I think that it is not very useful. In an attempt to
make it a bit simpler (one doesn't have to worry about the possibility
that a = 0 or b = 0), I used the parametric form of a line through
points (a1,a2) and (b1,b2). The form is

x = a1+t(b1-a1)
y = a2+t(b2-a2).

Replace x and y in the circle equation (x-h)^2+(y-k)^2-r^2 = 0 by the
expressions x = a1+t(b1-a1) and y = a2+t(b2-a2) and then solve for t.
In general, you'll obtain two values of t. Substitute these into the
parametric equations for the intersection points. You can look just
at the discriminant of the quadratic in the solution. If the
discriminant is positive, there are two intersection points; if the
discriminant is zero, just one (tangency); if the discriminant is
negative, there is no intersection.

t = (2*a1^2 + 2*a2^2 - 2*a1*b1 - 2*a2*b2 - 2*a1*h +
2*b1*h - 2*a2*k + 2*b2*k -
Sqrt[(-2*a1^2 - 2*a2^2 + 2*a1*b1 + 2*a2*b2 +
2*a1*h - 2*b1*h + 2*a2*k - 2*b2*k)^2 -
4*(a1^2 + a2^2 - 2*a1*b1 + b1^2 - 2*a2*b2 + b2^2)*
(a1^2 + a2^2 - 2*a1*h + h^2 - 2*a2*k + k^2 - r^2)])/
(2*(a1^2 + a2^2 - 2*a1*b1 + b1^2 - 2*a2*b2 + b2^2))}

t = (2*a1^2 + 2*a2^2 - 2*a1*b1 - 2*a2*b2 - 2*a1*h +
2*b1*h - 2*a2*k + 2*b2*k +
Sqrt[(-2*a1^2 - 2*a2^2 + 2*a1*b1 + 2*a2*b2 +
2*a1*h - 2*b1*h + 2*a2*k - 2*b2*k)^2 -
4*(a1^2 + a2^2 - 2*a1*b1 + b1^2 - 2*a2*b2 + b2^2)*
(a1^2 + a2^2 - 2*a1*h + h^2 - 2*a2*k + k^2 - r^2)])/
(2*(a1^2 + a2^2 - 2*a1*b1 + b1^2 - 2*a2*b2 + b2^2))

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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