$A$ focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$. Then the length of the semi-major axis is

The equation of the ellipse whose distance between the foci is $8$ and the distance between the directrices is $18$,is

Let the common tangents to the curves $4(x^{2}+y^{2}) = 9$ and $y^{2} = 4x$ intersect at the point $Q$. Let an ellipse,centered at the origin $O$,have 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 eccentric angle in the first quadrant of a point on the ellipse $\frac{x^{2}}{10}+\frac{y^{2}}{8}=1$ at a distance $3$ units from the centre of the ellipse is

The angle between the tangents drawn from a point $(-3, 2)$ to the ellipse $4x^2 + 9y^2 - 36 = 0$ is

If the foci of an ellipse are $(\pm \sqrt{5}, 0)$ and its eccentricity is $\frac{\sqrt{5}}{3}$,then the equation of the ellipse is