Tangent Common to Two Ellipses

Date: 10/24/2002 at 05:53:10
From: Amyn Poonawala
Subject: Common tangent to 2 ellipses

Hello Dr. Math,

I have two general ellipses in space and I want to find the equation 
of the tangent common to these ellipses. The two ellipses do not 
intersect each other and are not enclosed within each other. Hence, I 
believe there will be four such lines that will be common tangents. 
The question is, how do I find the equations of these lines?

Thanks,
Amyn

Date: 10/24/2002 at 08:29:17
From: Doctor Mitteldorf
Subject: Re: Common tangent to 2 ellipses

Dear Amyn,

The straightforward approach is to solve 4 equations in 4 unknowns
to find the two points (x1,y1) and (x2,y2) (on the two ellipses) that
are linked. Here is how you can get your 4 equations:

1) Two of the equations are the known equations for the ellipses, and
   they connect x1 to y1, and connect x2 to y2.

2) From the equations for the ellipses, derive the slope at any point.
   You can do this with "implicit differentiation."  For example, 

 if  x^2/a^2 + y^2/b^2 = 1, then

     2x dx        2y dy
   ---------- + ---------- = 0
       a^2          b^2 

                 -x*b^2
so that dy/dx = ---------
                  y*a^2 

Two more equations come from (a) setting these two slopes equal to
each other, and (b) setting them equal to the slope of the line
connecting the two points

                    y2-y1
                   -------
                    x2-x1

You still have a mess of algebra to do in combining these 4 equations.
If you make progress simplifying the equations, will you write back
and let me know what you find?

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