If distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is

Let the line $2 \mathrm{x}+3 \mathrm{y}-\mathrm{k}=0, \mathrm{k}>0$, intersect the $\mathrm{x}$-axis and $\mathrm{y}$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $\mathrm{AB}$ as a diameter is $\mathrm{x}^2+\mathrm{y}^2-3 \mathrm{x}-2 \mathrm{y}=0$ and the length of the latus rectum of the ellipse $\mathrm{x}^2+9 \mathrm{y}^2=\mathrm{k}^2$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\mathrm{m}$ and $\mathrm{n}$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

For the ellipse $3{x^2} + 4{y^2} = 12$, the length of latus rectum is

For the ellipse $25{x^2} + 9{y^2} - 150x - 90y + 225 = 0$ the eccentricity $e = $

Tangents at extremities of latus rectum of ellipse $3x^2 + 4y^2 = 12$ form a rhombus of area (in $sq.\ units$) -

The eccentricity of an ellipse whose length of latus rectum is equal to distance between its foci, is