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 \(ab?\)

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 \(k?\)

Find the equation of the tangent line to the ellipse \(1x^2+y^2=53\) at the point \((2,7)\) on the ellipse.

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.

Find the maximum area of the rectangle which is inscribed in the ellipse \[\frac{x^{2}}{5^{2}}+\frac{y^{2}}{2^{2}} = 1.\]