The product of perpendiculars from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ on the tangent at any point on the ellipse is

If $A_1, A_2, A_3$ are the areas of ellipse $x^2+4 y^2-4=0$ its director circle and auxiliary circle respectively, then $A_2+A_3-A_1=$

If the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ having $(1,1)$ as its middle point is $x+\alpha y=\beta$, then

Let F and $F^1$ be the foci of the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1(b<2)$ and $B$ is one end of the minor axis. If the area of the triangle $\mathrm{FBF}^1$ is $\sqrt{3}$ sq units, then the eccentricity of the ellipse is