The product of the lengths of perpendiculars from the foci on any tangent to the ellipse $3x^2 + 5y^2 = 1$ is:

Let $f(x)=x^2+9$,$g(x)=\frac{x}{x-9}$,$a=f(g(10))$,and $b=g(f(3))$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$,then $8e^2+l^2$ is equal to.

If the line $\frac{x}{a} + \frac{y}{b} = \sqrt{2}$ is tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then its eccentric angle $\theta = ............^{\circ}$

Find the equation of the ellipse whose center is at $(2, -3)$,focus is at $(3, -3)$,and one vertex is at $(4, -3)$.

For the ellipse $4x^2 + y^2 - 8x + 2y + 1 = 0$,the focus and the equation of the directrix are respectively

On the ellipse $\frac{x^2}{18} + \frac{y^2}{8} = 1$,the point $M$ nearest to the line $2x - 3y + 25 = 0$ is