Challenging Analytic Geometry Problem

Two congruent black circles are placed diagonally in a unit square such that they touch each other at the center of the square. We want to place two congruent ellipses (red) along the other diagonal such that they touch the sides of the square and the two black circles as shown in the figure above. The major axis of each of the two ellipses is taken along the diagonal parallel to the line $y = x$ while the minor axis is parallel to the line $y = - x$ as shown in the figure above. If the ratio of the semi-minor axis length $b$ to the semi-major axis length $a$ is $\dfrac{1}{2}$ , find the semi-major axis length $a$ . As your answer enter $\lfloor 10^4 a \rfloor$