The points of intersection of the perpendicular tangents drawn to the ellipse $4x^2 + 9y^2 = 36$ lie on the curve

In a group of $100$ persons,$75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$,then the eccentricity of the ellipse $25(\beta^2 x^2 + \alpha^2 y^2) = \alpha^2 \beta^2$ is $.......$

The equation of an ellipse whose eccentricity is $1/2$ and the vertices are $(4, 0)$ and $(10, 0)$ is

If the lines joining the foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$) to an extremity of its minor axis are inclined at an angle of $60^{\circ}$ to each other, then the eccentricity of the ellipse is:

Let the line $y-x=1$ intersect the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{1}=1$ at the points $A$ and $B$. Then the angle subtended by the line segment $AB$ at the center of the ellipse is:

If the line $\alpha x+4y=\sqrt{7}$, where $\alpha \in R$, touches the ellipse $3x^{2}+4y^{2}=1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is: