If the shortest distance between the lines $$\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$$ and $$\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$$ is $$\frac{44}{\sqrt{30}}$$, then the largest possible value of $$|\lambda|$$ is equal to _________.

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and $$x= \pm \frac{4}{\sqrt{3}}$$, respectively. Let the line $$y-\sqrt{3} x+\sqrt{3}=0$$ touch this hyperbola at $$\left(x_0, y_0\right)$$. If $$\mathrm{m}$$ is the product of the focal distances of the point $$\left(x_0, y_0\right)$$, then $$4 \mathrm{e}^2+\mathrm{m}$$ is equal to _________.

Let $$\alpha, \beta$$ be roots of $$x^2+\sqrt{2} x-8=0$$. If $$\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$$, then $$\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}$$ is equal to ________.

If the system of equations

$$\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$$

has infinitely many solutions, then $$(\lambda-\mu)$$ is equal to ______ :