if I know two ellipse, how could I get the common tangent between them?

by algebra, I got this equations
already(http://mathforum.org/library/drmath/view/61599.html):

a1*x1*x1+b1*x1*y1+c1*y1*y1+d1*x1+e1*y1+f1=0  (x1,y1) is on ellipse 1
a2*x2*x2+b2*x2*y2+c2*y2*y2+d2*x2+e2*y2+f2=0  (x2,y2) is on ellipse 2

(2*a1*x1+b1*y1+d1)/(b1*x1+2c1y1+e1) = (y2-y1)/x2-x1)  the slope of
tangent at point(x1,y1) equals the slop of the common tangent

(2*a2*x2+b2*y2+d2)/(b2*x2+2c2y2+e2) = (y2-y1)/x2-x1)  the slope of
tangent at point(x2,y2) equals the slop of the common tangent

based on those four euqations, we should be able to get the solutions
(x1,y1)(x2,y2)

but does anybody know how to get it and maybe in a computer program
how to formalize it to code? or get the approximate solutions by
iterations. 
I know it's kind of difficult to solve this problem, I would like to
pay more based on the speed and degree of the your work. thanks!