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

Consider ellipses $E_{k}: kx^{2} + k^{2}y^{2} = 1$,for $k = 1, 2, \ldots, 20$. Let $C_{k}$ be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse $E_{k}$. If $r_{k}$ is the radius of the circle $C_{k}$,then the value of $\sum_{k=1}^{20} \frac{1}{r_{k}^{2}}$ is $.......$.

The eccentricity of the ellipse $4x^2 + 9y^2 = 36$ is

The coordinates of the foci of the ellipse $16x^{2} + 9y^{2} = 144$ are

An ellipse is described by using an endless string which is passed over two pins. If the axes are $6 \ cm$ and $4 \ cm$,the necessary length of the string and the distance between the pins respectively in $cm$,are

The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r, s)$. Then the average of $\cos(\theta_1-\theta_2)$,$\cos(\theta_2-\theta_3)$ and $\cos(\theta_3-\theta_1)$ is