If the eccentricity and the length of the latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are $\frac{\sqrt{3}}{2}$ and $1$ respectively,then the sum of the lengths of the major axis and minor axis of the ellipse is

Consider an ellipse,whose centre is at the origin and its major axis is along the $x-$ axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is $6$,then the area (in sq. units) of the quadrilateral inscribed in the ellipse,with the vertices as the vertices of the ellipse,is

The equation of the ellipse whose foci are $(\pm 5, 0)$ and one of its directrices is $5x = 36$ is:

The equation of the tangent of the ellipse $4x^2 + 9y^2 = 36$ at the end of the latus rectum lying in the second quadrant is:

The eccentricity of the ellipse $4x^2 + y^2 - 8x + 2y + 1 = 0$ is

In an ellipse,the distance from one of the foci to its corresponding end of the major axis is $4-\sqrt{7}$ and the distance from the same focus to one end of the minor axis is $4$. Then the cosine of the angle subtended by the line segment joining its foci at one end of its minor axis is