If $A \equiv (x, 4, -1)$,$B \equiv (3, x, -5)$,and $C \equiv (2, -2, 3)$ are the vertices and $G \equiv (2, 1, -1)$ is the centroid of the triangle $ABC$,then the value of $x$ is

Let $A(x, y, z)$ be a point in the $xy$-plane,which is equidistant from three points $P(0, 3, 2)$,$Q(2, 0, 3)$,and $R(0, 0, 1)$. Let $B = (1, 4, -1)$ and $C = (2, 0, -2)$. Then among the statements $(S1) :$ $\triangle ABC$ is an isosceles right-angled triangle and $(S2) :$ the area of $\triangle ABC$ is $\frac{9 \sqrt{2}}{2}$.

The distance between the orthocentre and circumcentre of the triangle formed by the points $(1,2,3), (3,-1,5)$ and $(4,0,-3)$ is

$\triangle ABC$ has vertices at $A \equiv (2, 3, 5)$,$B \equiv (-1, 3, 2)$ and $C \equiv (\lambda, 5, \mu)$. If the median through $A$ is equally inclined to the axes,then the values of $\lambda$ and $\mu$ respectively are

The four points $A(2,-1,3), B(4,-2,1), C(4,5,-7)$ and $D(2,6,-5)$ form a:

If the points $A(1, 3, 5)$, $B(2, 4, 6)$, and $C(4, 5, k)$ form a right-angled triangle, then the number of possible values of $k$ is: