Basics of 3D Geometry

If a point $R\left(4,y,z\right)$ lies on the line segment joining the points $P\left(2,-\mathrm{3,4}\right)$ and $Q\left(\mathrm{8,0},10\right),$ then the distance of $R$ from the origin is

The plane which bisects the line segment joining the points $\left(-3, -3, 4\right)$ and $\left(3, 7, 6\right)$ at right angles, passes through which one of the following points?

A variable plane passes through a fixed point $\left(3, 2, 1\right)$ and meets $x, y$ and $z$-axes at $A, B \& C$ respectively. A plane is drawn parallel to the $yz–$ plane through $A$, a second plane is drawn parallel to the $zx-$ plane through $B$ and a third plane is drawn parallel to the $xy$- plane through $C$. Then the locus of the point of intersection of these three planes, is

If a variable plane, at a distance of $3$ units from the origin, intersects the coordinate axes at $A, B \& C$, then the locus of the centroid of $\mathrm{\Delta }ABC$ is

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A, B, C$ and $P$ are $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{4}$ respectively, then the position vector of the orthocentre of this triangle, is :