General Equation for Intersections of Line and Ellipse

Date: 03/13/2007 at 06:04:49
From: Baris
Subject: Intersection of line and ellipse

I am trying to find the intersection points (if there are any) of a
line and ellipse.  The line is defined by its 2 points, say (x0,y0)
and (x1,y1), and the ellipse is defined as x^2/a^2 + y^2/b^2 = r^2
(center is at the origin).  Is there a simple way to find any
intersections?

First I wrote the line's equation:

y-y0 = ((y1-y0)/(x1-x0))(x-x0)
   y = ((y1-y0)/(x1-x0))(x-x0) + y0

then I tried to put y into the ellipse equation so I can have a second
degree equation with x as the only variable.  But the equation becomes
too complicated.

Date: 03/13/2007 at 06:35:10
From: Doctor George
Subject: Re: Intersection of line and ellipse

Hi Baris,

Thanks for writing to Doctor Math.

It looks like you were doing the right things.  See if this approach
makes the algebra a little simpler.

Use your two points to construct the parametric equations for the line
as follows.

        x = x0 + at
        y = y0 + bt

Now substitute these expressions for x and y into the ellipse equation
and solve for t.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/