If the circles x 2 + y 2 + 5 K x + 2 y + K = 0 and 2 x 2 + y 2 + 2 K x + 3 y - 1 = 0 , K ∈ R ,…

$S_1=x^2+y^2+5Kx+2y+K=0$

$S_2=x^2+y^2+Kx+\frac{y}{2}+\frac{1}{2}=0$

$\therefore$ Equation of common chord is $S_1-S_2=0$

$\Rightarrow 4Kx+\frac{3}{2}y+\left(K-\frac{1}{2}\right)=0 ....\left(1\right)$ 
Give equation of common chord is $4x-5y-K=0 ....\left(2\right)$

$\because \left(1\right) \&\hspace{0.17em}\left(2\right)$ are identical
$\therefore \frac{4K}{4}=\frac{{\displaystyle \frac{3}{2}}}{-5}=\frac{K-\frac{1}{2}}{-K}$

Clearly, no value of $K$ satisfy simultaneously.