Thank you for reporting, we will resolve it shortly
Q.
If the circles $x^2 + y^2 + 5Kx + 2y + K = 0 $ and $2(x^2 + y^2) + 2Kx + 3y -1 = 0, (K \in R) $ , intersect at the points $P$ and $Q$, then the line $4x + 5y - K = 0$ passes through $P$ and $Q$ for :
Equation of common chord
$4kx + \frac{1}{2} y + k + \frac{1}{2} = 0 $ ....(1)
and given line is 4x + 5y -k = 0 .....(2)
On comparing (1) & (2), we get
$k = \frac{1}{10} = \frac{ k + \frac{1}{2}}{-k}$
$\Rightarrow $ No real value of $k$ exist