Let a line $L_1$ pass through the origin and be perpendicular to the lines

$\mathrm{L}_2: \overrightarrow{\mathrm{r}}=(3+\mathrm{t}) \hat{i}+(2 \mathrm{t}-1) \hat{j}+(2 \mathrm{t}+4) \hat{k}$ and

$\mathrm{L}_3: \overrightarrow{\mathrm{r}}=(3+2 \mathrm{~s}) \hat{i}+(3+2 \mathrm{~s}) \hat{j}+(2+\mathrm{s}) \hat{k}, \mathrm{t}, \mathrm{s} \in \mathbf{R}$.

If $(a, b, c), a \in \mathbf{Z}$, is the point on $\mathrm{L}_3$ at a distance of $\sqrt{17}$ from the point of intersection of $\mathrm{L}_1$ and $\mathrm{L}_2$, then $(\mathrm{a}+\mathrm{b}+\mathrm{c})^2$ is equal to $\_\_\_\_$ .

Consider the circle C : $x^2+y^2-6 x-8 y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x-\beta y-\gamma=0$, then $\alpha+\beta+2 \gamma$ is equal to $\_\_\_\_$ .

Let $f$ be a polynomial function such that $\log _2(f(x))=\left(\log _2\left(2+\frac{2}{3}+\frac{2}{9}+\ldots \ldots \infty\right)\right) \cdot \log _3\left(1+\frac{f(x)}{f(1 / x)}\right), x>0$ and $f(6)=37$. Then $\sum\limits_{\mathrm{n}=1}^{10} f(\mathrm{n})$ is equal to $\_\_\_\_$ .

A new unit ( $\alpha$ ) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of $\alpha$ units if light takes 6 min. 40 s to cover this distance?