If the circles `x^(2) + y^(2) + 5 Kx + 2y + K = 0` and `2x^(2) + y^(2)) + 2Kx + 3y - 1 = 0, (K in R)` intersect at the point P and Q then the line `4x + 5y - K = 0` passes P and Q for :

Options

A. exactly one value of K
B. infinitely many values of K
C. no values of K
D. exactly two values of K

Explanation

To solve the problem, we need to find the values of \( K \) such that the line \( 4x + 5y - K = 0 \) passes through the intersection points \( P \) and \( Q \) of the two given circles.

### Step-by-Step Solution:

1. **Identify the equations of the circles:**
The equations of the circles are:
\[ S_1: x^2 + y^2 + 5Kx + 2y + K = 0 \]
\[ S_2: 2x^2 + y^2 + 2Kx + 3y - 1 = 0 \]

2. **Find the equation of the radical axis (common chord):**
The radical axis can be found by subtracting the two circle equations:
\[ S_1 - S_2 = 0 \]
This gives:
\[ (x^2 + y^2 + 5Kx + 2y + K) - (2x^2 + y^2 + 2Kx + 3y - 1) = 0 \]
Simplifying this:
\[ -x^2 + 3Kx - y + K + 1 = 0 \]
Rearranging:
\[ x^2 - 3Kx + y - (K + 1) = 0 \]

3. **Set the radical axis equal to the line equation:**
The line equation is:
\[ 4x + 5y - K = 0 \]
We want this line to be identical to the radical axis. Thus, we can equate the coefficients:
\[ 4x + 5y - K = 0 \quad \text{and} \quad x^2 - 3Kx + y - (K + 1) = 0 \]

4. **Compare coefficients:**
From the radical axis equation, we can express it in the form:
\[ y = -\frac{1}{5} (4x - K) \]
By comparing coefficients, we have:
- Coefficient of \( x \): \( 4 = -3K \)
- Coefficient of \( y \): \( 5 = 1 \)
- Constant term: \( -K = -(K + 1) \)

5. **Solve for \( K \):**
From \( 4 = -3K \):
\[ K = -\frac{4}{3} \]
From the constant term:
\[ -K = -K - 1 \implies 0 = -1 \quad \text{(which is not possible)} \]

6. **Check for consistency:**
The equations derived from the coefficients must hold true. If they lead to contradictions, it implies that there are no values of \( K \) for which the line passes through both intersection points.

### Conclusion:
After analyzing the equations and finding contradictions, we conclude that there are no values of \( K \) for which the line \( 4x + 5y - K = 0 \) passes through both intersection points \( P \) and \( Q \).