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Intercept of a Line with a Circle


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
                   r: Circle's Radius

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.


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/   
    
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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