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 $.......$

Assertion $(A)$: If the tangent and normal to the ellipse $9x^2 + 16y^2 = 144$ at the point $P(\frac{\pi}{3})$ on it meet the major axis in $Q$ and $R$ respectively,then $QR = \frac{57}{8}$.
Reason $(R)$: If the tangent and normal to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $P(\theta)$ on it meet the major axis in $Q$ and $R$ respectively,then $QR = \left| \frac{a^2 \sin^2 \theta + b^2 \cos^2 \theta}{a \cos \theta} \right|$.

The area of the region bounded by the curve $x = 4 \cos \theta, y = 3 \sin \theta$ is . . . . . . sq. units. (in $\pi$)

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

The foci of $16x^2 + 25y^2 = 400$ are

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