If the lines joining the foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$) to an extremity of its minor axis are inclined at an angle of $60^{\circ}$ to each other, then the eccentricity of the ellipse is:

If a tangent having slope $m = \frac{1}{3}$ to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ is a normal to the circle $(x + 1)^2 + (y + 1)^2 = 1$,then $a^2$ lies in the interval:

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

$a$ and $b$ are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes. If its latus rectum is of length $4$ units and the distance between its foci is $4 \sqrt{2}$,then $a^2+b^2=$

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

If the area of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{\lambda^{2}}=1$ is $20 \pi$ sq units,then $\lambda$ is