Sticky Problem

king_of_inismac

Hey guys,

Need a bit of help with a seemingly simple problem.

I have an overlapping circle and ellipse. I have the equation for both.

The centre of the circle is say (5,5) and the foci of the ellipse are say (5,5) and (10,5), so one of the foci is the centre of the circle.

I need to find the equation of the 2 common tangents (and consequently, the intersection points of the tangents with the circle and ellipse).

There's a similar question here: http://answers.google.com/answers/threadview?id=401752

and the bones of a solution here:
http://mathforum.org/library/drmath/view/61599.html

Can anyone give me some help/advice?

M

RoundTower

I would start by finding the generalized equation for the set of all lines tangent to the circle. For example, for a circle centred at the origin, the tangent line at the point (x1, y1) has slope -x1/y1, so you can find the equation easily, it will be a quadratic in terms of x (you can substitute y = r^2 - x^2). But I think better than this would be to take a parameterized equation for the circle, i.e. replace x by cos t and y by sin t.

Then do the same for the ellipse (maybe it's more difficult).

So you end up with two equations like y = f(t)x + g(t) and y = h(s)x + i(s) where (I think) all the coefficients are linear combinations of cos t and sin t. For two lines to be identical, f(t) = h(s) and g(t) = i(s), so you have two simultaneous equations in s and t, and you will get either 0 or 2 solutions depending whether the ellipse is inside or outside the circle, or overlapping it.

To make the whole thing easier I'd let the centre of the circle be at (0,0) and move it back at the end.