The number of values of $k$ for which the points $A(-4, 9, k)$, $B(-1, 6, k)$, and $C(0, 7, 10)$ form a right-angled isosceles triangle is:

The coordinates of the foot of the perpendicular drawn from point $P(1, 0, 3)$ to the line joining points $A(4, 7, 1)$ and $B(3, 5, 3)$ are:

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 the orthocentre and the centroid of a triangle are at $(5,2,-6)$ and $(9,6,-4)$ respectively,then its circumcentre is

Let $A=(1,2,0)$, $B=(2,0,-1)$, $C=(0,-2,3)$ and $D=(-1,2,-3)$ be four points in space. Let $G_1$ be the centroid of triangle $ABC$ and $G_2$ be the centroid of tetrahedron $ABCD$. If $P$ divides $G_1G_2$ in the ratio $4:3$ internally, then $P=$

The incenter of the triangle $ABC$,whose vertices are $A(0,2,1)$,$B(-2,0,0)$,and $C(-2,0,2)$ is