The transformed equation of $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$ when the axes are translated to the point $(2, -3, 1)$ is

If $A(0,1,2)$,$B(2,-1,3)$,and $C(1,-3,1)$ are the vertices of a triangle,then the distance between its circumcentre and orthocentre is

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

If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then find the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta$.

$A(1,1,1), B(1,-4,3), C(2,-2,0)$ and $D(8,1,4)$ are the vertices of a tetrahedron. $G_1, G_2, G_3$ and $G_4$ are the centroids of the faces $ABC, BCD, CDA$ and $DAB$. Then the centroid of the tetrahedron having $G_1, G_2, G_3, G_4$ as its vertices is

The angle between two diagonals of a cube is