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

Let a tangent be drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in\left(0, \frac{\pi}{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to ..... .

If $\alpha, \beta$ are the eccentric angles of the extremities of a focal chord (other than the major axis) of the ellipse $x^2+4y^2=4$,then $\sqrt{3} \cos \frac{\alpha+\beta}{2} =$

If $S$ and $S'$ are the two foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a < b$,and $P(x_1, y_1)$ is a point on the ellipse,then $SP + S'P = \dots$

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 locus of the point of intersection of two perpendicular tangents to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is: