The point in the $xy$-plane which is equidistant from the points $A(2,0,3)$,$B(0,3,2)$,and $C(0,0,1)$ has the coordinates

Assertion $(A):$ If $(-1,3,2)$ and $(5,3,2)$ are respectively the orthocentre and circumcentre of a triangle, then $(3,3,2)$ is its centroid.
Reason $(R):$ Centroid of the triangle divides the line segment joining the orthocentre and the circumcentre in the ratio $1: 2$.
Which one of the following is true?

If $\left(\frac{9}{4}, \frac{5}{4}, \frac{15}{4}\right)$ is the centroid of a tetrahedron whose vertices are $(a, 2, 1), (1, b, 4), (4, 0, c)$ and $(1, 1, 7)$, then

The edge of a cube is of length $a$. Then the shortest distance between the diagonal of a cube and an edge skew to it is:

$A$ tetrahedron of volume $5$ has three of its vertices at the points $A(2,1,-1)$,$B(3,0,1)$,and $C(2,-1,3)$. If the fourth vertex $D$ lies on the $y$-axis,then the sum of the ordinates of all possible points $D$ is-

Find the distance of the point $P(-\hat{i} + 2\hat{j} + 6\hat{k})$ from the line passing through the point $A(2, 3, -4)$ and parallel to the vector $\vec{v} = 6\hat{i} + 3\hat{j} - 4\hat{k}$.