The locus of a variable point whose distance from $(-2, 0)$ is $\frac{2}{3}$ times its distance from the line $x = -\frac{9}{2}$ is

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

The eccentricity of an ellipse,with its centre as origin,is $1/2$. If one of the directrices is $x=4$,then the equation of the ellipse is given by

If a number of ellipses are described having the same major axis $2a$ but a variable minor axis,then the tangents at the ends of their latera recta pass through fixed points which are:

Find the coordinates of the foci,the vertices,the length of the major axis,the minor axis,the eccentricity,and the length of the latus rectum of the ellipse $\frac{x^{2}}{100}+\frac{y^{2}}{400}=1$.

Let $L$ be a common tangent line to the curves $4x^{2} + 9y^{2} = 36$ and $(2x)^{2} + (2y)^{2} = 31$. Then the square of the slope of the line $L$ is ..... .