If the distance of point $B(2, 1, -3)$ from the line passing through the point $A(4, -2, 2)$ and parallel to the vector $\vec{c} = -4\hat{i} - 6\hat{j} - 2\hat{k}$ is $x$, then $x^4 + x^2 + 541 =$

What is the equation of the line passing through the point $(a, b, c)$ and parallel to the $z$-axis?

If a point $P$ on the line segment joining the points $(3, 5, -1)$ and $(6, 3, -2)$ has its $y$-coordinate $2$,then its $z$-coordinate is

If the line joining the points $(k, 2, 3)$ and $(1, 1, 2)$ is parallel to the line joining the points $(5, 4, -1)$ and $(3, 2, -3)$,then the value of $k$ is equal to

If the shortest distance between the lines $\vec{r}_{1}=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0$ and $\vec{r}_{2}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R$ is $9$,then $\alpha$ is equal to $.....$

On a line with direction cosines $l, m, n$,$A(x_1, y_1, z_1)$ is a fixed point. If $B = (x_1 + 4kl, y_1 + 4km, z_1 + 4kn)$ and $C = (x_1 + kl, y_1 + km, z_1 + kn)$ where $k > 0$,then the ratio in which the point $B$ divides the line segment joining $A$ and $C$ is: