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

The equation of the ellipse with directrix $3 x+4 y-5=0$, focus $(1,2)$ and eccentricity $1 / 2$, is

The ellipse having its foci $(0, \pm 1)$ and major axis of length $\sqrt{5}$ is