If $\pi / 3, \theta$ are the eccentric angles of the ends of a focal chord of the ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$, then $\tan \theta=$

If $x+2 y+k=0, k>0$ is a tangent to the ellipse $2 x^2+y^2=2$, then the equation of the normal to the given ellipse at $\left(\frac{1}{\sqrt{2}}, \frac{k}{3}\right)$, is

If $A=(1,2), B=(2,1)$ and $P$ is any point satisfying the condition $P A+P B=3$, then the equation of the locus of $P$ is

If the sum of the distances from the foci to the centre $O(0,0)$ of an ellipse is $8 \sqrt{6}$ units and the area of the smallest rectangle in which that ellipse is inscribed is 80 sq. units, then the equation of such an ellipse is