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 an ellipse whose eccentricity is $1/2$ and the vertices are $(4, 0)$ and $(10, 0)$ is

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be an ellipse with foci $F_1$ and $F_2$. Let $AO$ be its semi-minor axis, where $O$ is the centre of the ellipse. The lines $A F_1$ and $A F_2$, when extended, cut the ellipse again at points $B$ and $C$ respectively. Suppose that the $\triangle A B C$ is equilateral. Then, the eccentricity of the ellipse is

If $P \equiv (x,\;y)$, ${F_1} \equiv (3,\;0)$, ${F_2} \equiv ( - 3,\;0)$ and $16{x^2} + 25{y^2} = 400$, then $P{F_1} + P{F_2}$ equals

The equation of ellipse whose distance between the foci is equal to $8$ and distance between the directrix is $18$, is