The length of the latus rectum of an ellipse is $\frac{18}{5}$ and eccentricity is $\frac{4}{5}$,then the equation of the ellipse is...

Let the eccentricity of the ellipse $2x^2 + ay^2 - 8x - 2ay + (8 - a) = 0$ be $\frac{1}{\sqrt{3}}$. If the major axis of this ellipse is parallel to the $Y$-axis,then the equation of the tangent to this ellipse with slope $1$ is:

The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon $P$ of $n$ sides and $q$ be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p+q=126$,then the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{n}=1$ is :

If the chord through the points whose eccentric angles are $\theta$ and $\phi$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passes through the focus,then the value of $(1 + e) \tan(\frac{\theta}{2}) \tan(\frac{\phi}{2})$ is

The length of the latus rectum of the ellipse $5x^2 + 9y^2 = 45$ is