In the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x>0$, if the term independent of $x$ is (221)k, then k is equal to:

Let $\mathrm{P}(3 \cos \alpha, 2 \sin \alpha), \alpha \neq 0$, be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1, \mathrm{Q}$ be a point on the circle $x^2+y^2-14 x-14 y+82=0$ and R be a point on the line $x+y=5$ such that the centroid of the triangle PQR is $\left(2+\cos \alpha, 3+\frac{2}{3} \sin \alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let $\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\frac{8}{3}$. If the line $x=\alpha$ intersects the hyperbola H at the points A and B such that the area of the triangle AOB is $4 \sqrt{15}$, where O is the origin, then $\alpha^2$ equals

The shortest distance between the lines

$$ \vec{r}=\left(\frac{1}{3} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\frac{8}{3} \hat{\mathrm{k}}\right)+\lambda(2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}) $$

and $\vec{r}=\left(-\frac{2}{3} \hat{\mathrm{i}}-\frac{1}{3} \hat{\mathrm{k}}\right)+\mu(\hat{\mathrm{j}}-\hat{\mathrm{k}}), \lambda, \mu \in \mathbb{R}$, is: