If the circles x ^ 2 + y ^ 2 + 5Kx + 2y + K = 0 and 2( x ^ 2 + y ^...

${S}_{1} = {x}^{2} + {y}^{2} + 5Kx + 2y + K = 0$

${S}_{2} = {x}^{2} + {y}^{2} + Kx + \dfrac{y}{2} + \dfrac{1}{2} = 0$

$∴$ Equation of common chord is ${S}_{1} - {S}_{2} = 0$

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

$∵ \left(1\right) \text{ and } \left(2\right)$ are identical
$∴ \dfrac{4K}{4} = \dfrac{\dfrac{3}{2}}{- 5} = \dfrac{K - \dfrac{1}{2}}{- K}$

Clearly, no value of $K$ satisfy simultaneously.