The mid point of the line segment joining the points $A\left(p,-4.2\right)$ and $B\left(3,2,-4\right)$  is given by $M\left(\frac{p+3}{2},-1,-1\right)$

Now given $\left(2,3,c\right)$ is direction ratio of line which passing through $C\left(5,q,1\right) \& M\left(\frac{p+3}{2},-1,-1\right)$

So, by direction ratio formula between two points we get,

$\frac{p+3}{2}-5=2\Rightarrow p=11$

$-1-q=3\Rightarrow q=-4$

$-1-1=c\Rightarrow c=-2$

Now solving $c.\left(p+7q\right)$ by using above value we get,

$c.\left(p+7q\right)=-2\times \left(11-28\right)=34$