Ellipse - Coordinate Geometry - Mathematics Previous Year Questions

If the tangent at a point \(\left( {4\cos \phi ,{{16} \over {\sqrt {11} }}\sin \phi } \right)\) to the ellipse \(16{x^2} + 11{y^2} = 256\) is also a tangent to \({x^2} + {y^2} - 2x = 15\), then \(\phi\) equsls

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at points where it is intersected by the line $l x+m y+n=0$. The point of intersection of tangents at these points is

A rectangle is inscribed in an ellipse with the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

What is the maximum area of the rectangle that can be inscribed in the ellipse?