Every point on the line `2x+11y-5=0` is at equal distance from the lines `24x+7y=20` and `4x-3y=2`.

Answer

To solve the problem, we need to show that every point on the line \(2x + 11y - 5 = 0\) is at an equal distance from the lines \(24x + 7y - 20 = 0\) and \(4x - 3y - 2 = 0\).

### Step-by-Step Solution:

1. **Rewrite the equations of the lines**:
- The first line is already given as \(2x + 11y - 5 = 0\).
- The second line can be rewritten as \(24x + 7y - 20 = 0\).
- The third line can be rewritten as \(4x - 3y - 2 = 0\).

2. **Identify the coefficients**:
- For the line \(24x + 7y - 20 = 0\), the coefficients are \(A_1 = 24\), \(B_1 = 7\), and \(C_1 = -20\).
- For the line \(4x - 3y - 2 = 0\), the coefficients are \(A_2 = 4\), \(B_2 = -3\), and \(C_2 = -2\).

3. **Check the sign of \(A_1B_2 - A_2B_1\)**:
- Calculate \(A_1B_2 - A_2B_1 = 24 \cdot (-3) - 4 \cdot 7 = -72 - 28 = -100\).
- Since this value is negative, it indicates that the angle between the two lines is acute.

4. **Find the equation of the angle bisectors**:
- The formula for the angle bisectors between two lines \(A_1x + B_1y + C_1 = 0\) and \(A_2x + B_2y + C_2 = 0\) is given by:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

5. **Substituting the values**:
- For the acute angle bisector:
\[ \frac{24x + 7y - 20}{\sqrt{24^2 + 7^2}} = \frac{4x - 3y - 2}{\sqrt{4^2 + (-3)^2}} \]
- Simplifying the left side:
\[ \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]
- Simplifying the right side:
\[ \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

6. **Setting up the equation**:
- The equation becomes:
\[ \frac{24x + 7y - 20}{25} = \frac{4x - 3y - 2}{5} \]
- Cross-multiplying gives:
\[ 5(24x + 7y - 20) = 25(4x - 3y - 2) \]

7. **Expanding and simplifying**:
- Expanding both sides:
\[ 120x + 35y - 100 = 100x - 75y - 50 \]
- Rearranging gives:
\[ 120x - 100x + 35y + 75y - 100 + 50 = 0 \]
- This simplifies to:
\[ 20x + 110y - 50 = 0 \quad \text{or} \quad 2x + 11y - 5 = 0 \]

8. **Conclusion**:
- The equation \(2x + 11y - 5 = 0\) is indeed the angle bisector, confirming that every point on this line is equidistant from the two given lines.