Let $$\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha $$ and $$\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \beta $$.

If $$\int\limits_{\beta - {8 \over 3}}^{2\alpha - 1} {Max\left\{ {{{9 - {x^2}} \over {5 - x}},x} \right\}dx = {\alpha _1} + {\alpha _2}{{\log }_e}\left( {{8 \over {15}}} \right)} $$ then $${\alpha _1} + {\alpha _2}$$ is equal to _____________.

If two tangents drawn from a point ($$\alpha$$, $$\beta$$) lying on the ellipse 25x² + 4y² = 1 to the parabola y² = 4x are such that the slope of one tangent is four times the other, then the value of (10$$\alpha$$ + 5)² + (16$$\beta$$² + 50)² equals ___________.

Let S be the region bounded by the curves y = x³ and y² = x. The curve y = 2|x| divides S into two regions of areas R₁, R₂. If max {R₁, R₂} = R₂, then $${{{R_2}} \over {{R_1}}}$$ is equal to ______________.

If the shortest distance between the lines

$$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\widehat j} \right)$$

and $$\overrightarrow r = \left( { - \widehat j + 2\widehat k} \right) + \mu \left( {\widehat i - \widehat j + \widehat k} \right)$$ is $$\sqrt {{2 \over 3}} $$, then the integral value of a is equal to ___________.