If the line whose equation is `9x-2ky+k=0` passes through intersection of the lines whose equations are `2x+7y+22=0` and `5x-y+19=0` then the value of `|k|` is equal to

Answer

To solve the problem, we need to find the value of \(|k|\) given that the line \(9x - 2ky + k = 0\) passes through the intersection of the lines \(2x + 7y + 22 = 0\) and \(5x - y + 19 = 0\).

### Step-by-Step Solution:

1. **Find the intersection of the two lines \(L_1\) and \(L_2\)**:
- The equations of the lines are:
\[ L_1: 2x + 7y + 22 = 0 \]
\[ L_2: 5x - y + 19 = 0 \]
- We can solve these equations simultaneously to find their intersection point.

2. **Express \(y\) from \(L_2\)**:
- Rearranging \(L_2\):
\[ y = 5x + 19 \]

3. **Substitute \(y\) in \(L_1\)**:
- Substitute \(y\) in \(L_1\):
\[ 2x + 7(5x + 19) + 22 = 0 \]
- Simplifying:
\[ 2x + 35x + 133 + 22 = 0 \implies 37x + 155 = 0 \]
- Solving for \(x\):
\[ 37x = -155 \implies x = -\frac{155}{37} \]

4. **Find \(y\) using the value of \(x\)**:
- Substitute \(x\) back into the equation for \(y\):
\[ y = 5\left(-\frac{155}{37}\right) + 19 = -\frac{775}{37} + \frac{703}{37} = -\frac{72}{37} \]

5. **Intersection point**:
- The intersection point is:
\[ \left(-\frac{155}{37}, -\frac{72}{37}\right) \]

6. **Substituting the intersection point into \(L_3\)**:
- The line \(L_3\) is given by:
\[ 9x - 2ky + k = 0 \]
- Substitute \(x = -\frac{155}{37}\) and \(y = -\frac{72}{37}\):
\[ 9\left(-\frac{155}{37}\right) - 2k\left(-\frac{72}{37}\right) + k = 0 \]
- Simplifying:
\[ -\frac{1395}{37} + \frac{144k}{37} + k = 0 \]
- Multiplying through by \(37\) to eliminate the denominator:
\[ -1395 + 144k + 37k = 0 \implies 181k = 1395 \]
- Solving for \(k\):
\[ k = \frac{1395}{181} \]

7. **Finding \(|k|\)**:
- The value of \(|k|\) is:
\[ |k| = \frac{1395}{181} \]

### Final Answer:
\[ |k| = \frac{1395}{181} \approx 7.71 \]