1. Problem: ellipse x^2/a^2 + y^2/b^2 = 1 encloses circle x^2 + y^2 = 2x. Find values of a and b that minimize the area of the ellipse. 2. Relevant equations: A = pi*a*b for an ellipse. 3. The attempt at a solution: I tried a bunch of crazy stuff... I know I need to find where the tangents (1st derivatives) are equal for the ellipse and circle (at two intersections where x's are equal and y's will be mirrored across the x axis). Or I think I know that... I can do the implicit differentiation to get me the dy/dx for the ellipse and circle [-xb^2/ya^2 and (1-x)/y respectively] but I set them equal and don't really get anything out of it, so I tried solving the ellipse equation for y^2 [y^2=b^2/a^2(a^2-x^2)], plugging that into the circle equation, then getting dy/dx but that didn't seem to work either. I felt like I was making progress several times but now I just feel like I'm going in circles (or ellipses)... I can show much more of what I tried if it helps. I am still working on it as I post this. P.S. I just looked over an old (2008) thread here about a very similar problem which seems to be helping but I still can't quite get there with what I just read.