An ellipse with semi-major axis a and semi-minor axis b is sliding in place, such that it is always tangent to the x-axis at the origin. The center of the ellipse traces a closed locus. Find the area enclosed by this locus in terms of a and b.
In addition to the given ellipse E plot another ellipse E2 which is identical to E but always centered at the origin; E2 will always be a translation of E.
Now draw the upper horizontal tangent of E2 and trace the point of tangency as E2 is rotated. This will trace the same path as the center of E.
Let t be the angle of rotation. Then an equation covering all possible E2 can be written as (x*cos(t)-y*sin(t))^2/a^2 + (x*sin(t)+y*cos(t))^2/b^2 = 1.
Playing around with specific examples leads me to believe the locus is a circle with a diameter from (0,a) to (0,b). This circle can be written as x^2 + y^2 - (a+b)*y + ab = 0. Or using the same parameter t, it can be written as x=((a-b)/2)*sin(2t), y=((b-a)/2)*cos(2t) + (a+b)/2 which should satisfy the equation for E2.
Thoughts