Section Formula

The $XY$-plane divides the line joining the points $\left(-1, 3, 4\right), \left(2, -5, 6\right)$.

If $P,Q,R,S$ are four coplanar points on the sides$AB,BC,CD,DA$of a skew quadrilateral, then $\frac{AP}{PB}·\frac{BQ}{QC}·\frac{CR}{RD}·\frac{DS}{SA}$ equals

A plane $2x+3y+5z=1$ has a point $P$, which is at minimum distance from line joining $A\left(1,0,–3\right), B\left(1,–5,7\right)$, then distance $AP$ is equal to

In a three-dimensional coordinate system $P, Q, R$ are images of a point $A\left(a,b,c\right)$ in the $XY, YZ \& ZX$ planes respectively. If $G$ is the centroid of triangle $PQR$, then area of triangle $AOG$ is ($O$ is the origin)

$A\left(3,2,0\right), B\left(5,3,2\right)$  $C\left(–9,6,–3\right)$ are the vertices of a triangle $ABC$. If the bisector of $\angle ABC$ meets $BC$ at $D$, then coordinates of $D$ are

If $P\left(3,2,–4\right),Q\left(5,4,–6\right)$ and $R\left(\mathrm{9,}8,–10\right)$ are collinear, then $R$ divides $PQ$ in the ratio

The plane $XOZ$ divides the join of $\left(1,-1,5\right)$ and $\left(2,3,4\right)$ in the ratio $\lambda :1$, then $\lambda$ is

The ratio in which the $ZX$-plane divided the joining of $\left(1,2,3\right)$ and $\left(4,2,1\right)$.

Which of the following is/are correct about a tetrahedron?

If $A=\left(1,2,3\right),B=\left(4,5,6\right),C=\left(7,8,9\right)$ and $D,E,F$ are the midpoints of the triangle $ABC,$ then find the centroid of the$∆DEF$

If the mid-points of the sides of a triangle are $\left(1, 5, -1\right), \left(0, 4, -2\right)$ and $\left(2, 3, 4\right)$. Find its vertices.