If the area of the auxiliary circle of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\left( {a > b} \right)$ is twice the area of the ellipse, then the eccentricity of the ellipse is

If the foci and vertices of an ellipse be $( \pm 1,\;0)$ and $( \pm 2,\;0)$, then the minor axis of the ellipse is

Let the common tangents to the curves $4\left(x^{2}+y^{2}\right)=$ $9$ and $y ^{2}=4 x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and $6$ , respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac{l}{ e ^{2}}$ is equal to

The minimum area of a triangle formed by any tangent to the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{81}} = 1$ and the coordinate axes is 

The distance between the directrices of the ellipse $\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{20}} = 1$ is

Slope of common tangents of parabola $(x -1)^2 = 4(y -2)$ and ellipse ${\left( {x - 1} \right)^2} + \frac{{{{\left( {y - 2} \right)}^2}}}{2} = 1$ are $m_1$ and $m_2$ ,then $m_1^2 + m_2^2$ is equal to