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.
Find conditions on \(a > 0\) and \(b > 0\) such that the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is contained inside the square (and tangent to all of its sides).
I Suppose that the line \(\mathbf{y=x+ k}\) is tangent to the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\).
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