Let \((m,n)\) be the point at which a line is tangent to the ellipse \(\displaystyle{\frac{x^2}{16}+\frac{y^2}{100}=1}.\) If this line has \(x\)-intercept \(A=(a,0)\) and \(y\)-intercept \(B=(0,b),\) where \(a\) and \(b\) are positive, what is the minimum value of \(ab?\)

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The ellipse \(\displaystyle{\frac{x^2}{8}+\frac{y^2}{17}=1}\) and the line \(y=2x - k\) intersect at exactly two distinct points. What is the range of the constant \(k?\)

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In the above diagram, a rectangle \(ABCD\) is inscribed in the ellipse \(\displaystyle{\frac{x^2}{36}+\frac{y^2}{100}=1}.\) What is the maximum area of rectangle \( ABCD?\)

Note: The above diagram is not drawn to scale.

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