The locus of the point of intersection of the perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is

$P$ is a variable point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $AA'$ as the major axis. Then the maximum value of the area of $\Delta APA'$ is

If $P$ is any point on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ and $S$ and $S^{\prime}$ are the foci,then $PS + PS^{\prime}$ is equal to

The ellipse $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point $(0,4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E_2$ is

The equation of the ellipse with $x+y+2=0$ as its directrix,one of its focus at $(1,-1)$ and having eccentricity $e = \frac{2}{3}$ is:

The longest distance of the point $(a, 0)$ from the curve $2x^2+y^2=2x$ is