Ellipse

Find the co-ordinates of all the points $P$ on the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, for which the area of the triangle $PON$ is maximum, where $O$ denotes the origin and $N$, the foot of the perpendicular from $O$ to the tangent at $P$.

A tangent to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x² + 2y² = 6 are at right angles.

Let '$d$' be the perpendicular distance from the centre of the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ to the tangent drawn at a point $P$ on the ellipse. If ${F_1}$ and ${F_2}$ are the two foci of the ellipse, then show that ${\left( {P{F_1} - P{F_2}} \right)^2} = 4{a^2}\left( {1 - {{{b^2}} \over {{d^2}}}} \right)$.

An ellipse has eccentricity ${1 \over 2}$ and one focus at the point $P\left( {{1 \over 2},1} \right)$. Its one directrix is the common tangent, nearer to the point $P$, to the circle ${x^2} + {y^2} = 1$ and the hyperbol;a ${x^2} - {y^2} = 1$. The equation of the ellipse, in the standard form, is ............