Ellipse fit tangentally to a line modifying one axis

Question

A graphic of this problem is here:

http://dl.dropbox.com/u/13390614/Question.jpg

Take an axis aligned ellipse with a fixed minor axis, and stretch the ellipse along its major axis till it becomes tangental to some line segment (A in the graphic).

What are the coordinates of the tangental point (P), or, what would the major axis length be?

I know how to calc the major axis if I have the tangental point, and can calc the point if I have the major axis, but with neither, I'm stumped

I have also solved this when the minor axis is stretched along with the major, maintaining the ratio. The problem is when one axis is fixed.

Any insights would be appreciated, especially via trig.

Gary

Answer 1

Consider

x^2/max^2 + y^2/fix^2 = 1; % ellipse
Ax + By + C = 0;            % segment line

Then

x^2/max^2 + (Ax + C)^2/(B*fix)^2 = 1; // Quadratic equation

Your solution is when discriminant is equal to 0.

   x^2   (1/max^2 + A^2/(B*fix)^2)   
+  x      2 AC/(B*fix)^2
+        C^2/(B*fix)^2 - 1  
= 0

a = (1/max^2 + A^2/(B*fix)^2);
b = 2 AC/(B*fix)^2;
c = C^2/(B*fix)^2 - 1.

b^2 = 4ac   ==>   a = b^2/c    ==>
a = 4(AC)^2/(B*fix)^4 / ( C^2/(B*fix)^2 - 1 )
1/max^2 = 4(AC)^2/(B*fix)^4 / ( C^2/(B*fix)^2 - 1 ) - A^2/(B*fix)^2);