If the circles ${\mathrm{x}}^2+{\mathrm{y}}^2+5kx+2\mathrm{y}+\mathrm{k}=0$ and $2\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)+2kx+3\mathrm{y}-1=0,(\mathrm{k}\in \mathrm{R}),$ intersect at the points $P$ and $Q$, then the line $4\mathrm{x}+5\mathrm{y}-\mathrm{k}=0$ passes through $P$ and $Q$, for:

The given circles are $S_1=0,S_2=0$

Equation of common chord is ${\mathrm{S}}_1-{\mathrm{S}}_2=0$

                    $4kx+\frac{\mathrm{y}}{2}+\mathrm{k}+\frac{1}{2}=0\Rightarrow 8kx+y+2k+1=0$

Given that the eqution of the line passing through the points of intersection of circles is $4\mathrm{x}+5\mathrm{y}-\mathrm{K}=0$, it is common chord of the circles

Comparing the above two equations

                    $\frac{8k}{4}=\frac{1}{5}=\frac{2k+1}{-k}$

Consider the first part of the above equation 

                   $40k=4\Rightarrow k=\frac{1}{10}$

From second part of the equation 

                 $\begin{array}{rcl}\frac{2k+1}{-k}& =& \frac{1}{5}\\ 10k+5& =& -k\\ 11k& =& -5\\ k& =& -\frac{5}{11}\end{array}$

The above two values of $k$ are not same,

Therefore,  $\boxed{there is no value of k}$, 

for which $4\mathrm{x}+5\mathrm{y}-\mathrm{K}=0$ is line passing through the points of intersection of two circles given.