A Tangent Line Proportionality

The two foci of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) together with one endpoint of the minor axis are the vertices of a right triangle. The line $l:y=-x+3$ has exactly one common point $T$ with $E$.

1. Find the equation of $E$ and the coordinates of $T$. 2. Let $O$ be the origin. A line $l'$ parallel to $OT$ meets $E$ at two distinct points $A,B$ and meets $l$ at a point $P$. Prove that there is a constant $\lambda$ with $|PT|^2=\lambda|PA|\cdot|PB|$, and find $\lambda$.