$\begin{array}{1 1} (8,-10,2)\; and\; (6,-4,-2) \\ (8,10,2)\; and \;(6,-4,-2) \\(8,-10,2)\; and \;(6,4,-2) \\(8,-10,2)\; and \;(6,-4,2) \end{array} $

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- Section formula: The coordinates of the point $C$ that divides the segment joining the points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ in the ratio $l:m$ internally is given by $C\big(\large\frac{lx_2+mx_1}{l+m},\frac{ly_2+my_1}{l+m},\frac{lz_2+mz_1}{l+m}\big)$

Points of trisection are the points that divide $PQ$ in ratios $2:1$ and $1:2$

Let the points of tri sections of $PQ$ be $C$ and $D$

Step 1

Given points are$P(4,2,-6)$ and $ Q(10,-16,6)$

From section formula the coordinates of $C$ which divides $PQ$ in ratio $2:1$ is given by

$C\big(\large\frac{2.10+4}{2+1},\frac{2.(-16)+2}{2+1},\frac{2.6-6}{2+1}\big)$

$=C(8,-10,2)$

Step 2

From section formula the coordinates of $D$ which divides $PQ$ in ratio $1:2$ is given by

$D\big(\large\frac{2.4+10}{2+1},\frac{2.2-16}{2+1},\frac{2.(-6)+6}{2+1}\big)$

$=D(6,-4,-2)$

$\therefore$ The points of trisection are $(8,-10,2)$ and $(6,-4,-2)$

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