Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $\mathrm{x}$-axis. If its minor axis subtends an angle $60^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ____________.

If the sum of the series

$\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{2^{2}}-\frac{1}{2 \cdot 3}+\frac{1}{3^{2}}\right)+\left(\frac{1}{2^{3}}-\frac{1}{2^{2} \cdot 3}+\frac{1}{2 \cdot 3^{2}}-\frac{1}{3^{3}}\right)+$

$\left(\frac{1}{2^{4}}-\frac{1}{2^{3} \cdot 3}+\frac{1}{2^{2} \cdot 3^{2}}-\frac{1}{2 \cdot 3^{3}}+\frac{1}{3^{4}}\right)+\ldots$

is $\frac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are co-prime, then $\alpha+3 \beta$ is equal to __________.

If the area bounded by the curve $2 y^{2}=3 x$, lines $x+y=3, y=0$ and outside the circle $(x-3)^{2}+y^{2}=2$ is $\mathrm{A}$, then $4(\pi+4 A)$ is equal to ____________.

Let $f(x)=\int \frac{d x}{\left(3+4 x^{2}\right) \sqrt{4-3 x^{2}}},|x|<\frac{2}{\sqrt{3}}$. If $f(0)=0$

and $f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right)$, $\alpha, \beta>0$, then $\alpha^{2}+\beta^{2}$ is equal to ____________.