The eccentricity of the ellipse $x^2+4 y^2+2 x+16 y+13=0$ is

For an ellipse with eccentricity $e = \frac{1}{2}$,the centre is at the origin. If one of its directrices is $x = 4$,then the equation of the ellipse is

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

If lines $3x + 2y = 10$ and $-3x + 2y = 10$ are tangents at the extremities of the latus rectum of an ellipse whose centre is the origin,then the length of the latus rectum of the ellipse is:

If the straight line $x \cos \alpha + y \sin \alpha = p$ touches the curve $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,then prove that $a^{2} \cos^{2} \alpha + b^{2} \sin^{2} \alpha = p^{2}$.

The parametric form of a point on the ellipse whose foci are $(-1, 0)$ and $(7, 0)$ and eccentricity is $1/2$ is: