$P,Q,R,S$  are the points  $(1,2,-2)(8,10,11)(1,2,3)$ and  $(3,5,7)$ respectively.  If S  denotes the projection of PQ  on  RS then $29S^2+29=\_\_\_\_$

d.r’s of RS are  $(3-1,5-2,7-3)=(2,3,4)$
d.c’s of Rs  $=(\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}})$
Projection of PQ on RS is
$$\begin{gathered} S=\frac{1}{\sqrt{29}}[2(8-1)+3(10+2)+4(11+2)] \\ 29S^2={\left(90\right)}^2=8100 \\ 29S^2+29=8129 \end{gathered}$$