If the circles x² + y² + 52Kx + 2y + K = 0 and 2(x² + y²) + 2Kx + 3y - 1 = 0, (K ∈ R), intersect at the points P and Q. Then the line 4x + 5y - K = 0 passes through P and Q for:

The correct option is D no value of K
Equation of common chord 4Kx + 12y + k + 12 = 0.. (1)
and given line is 4x + 5y - k = 0.. (2)
On comparing (1) (2), we get
For the line 4x + 5y - K = 0 to pass through P and Q, it must be the equation of the common chord of the two circles.
The equation of the common chord of the circles x² + y² + 52Kx + 2y + K = 0 and 2(x² + y²) + 2Kx + 3y - 1 = 0 is given by subtracting the equations:
2(x² + y²) + 2Kx + 3y - 1 - 2(x² + y² + 52Kx + 2y + K) = 0
2x² + 2y² + 2Kx + 3y - 1 - 2x² - 2y² - 104Kx - 4y - 2K = 0
-102Kx - y - 1 - 2K = 0
102Kx + y + 1 + 2K = 0
Comparing this with 4x + 5y - K = 0, we must have:
102K = 4 (coefficient of x)
1 = 5 (coefficient of y) which is a contradiction.
1 + 2K = -K (constant term)
Therefore, there is no value of K for which the line 4x + 5y - K = 0 passes through P and Q.