Let k be a positive real number. The square with vertices (k,0), (0,k), (-k,0), and (0,-k) is plotted in the coordinate plane. It is possible to draw an ellipse so that it is tangent to all sides of the square; several examples are shown below.

Find necessary and sufficient conditions on a > 0 and b > 0 such that the ellipse
{x^2}/{a^2} + {y^2}/{b^2} = 1 is contained inside the square (and tangent to all of its sides).

Make sure to prove that your conditions are both necessary (the ellipse being tangent to all the square's sides implies your conditions) and sufficient (your conditions imply the ellipse being tangent to all the square's sides).