Find the distance between the parallel lines `5x+12y-20=0` and `5x+12y+6=0`.

Answer

To find the distance between the parallel lines given by the equations \(5x + 12y - 20 = 0\) and \(5x + 12y + 6 = 0\), we can use the formula for the distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\):

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

### Step-by-step solution:

1. **Identify the coefficients**:
From the equations of the lines, we can identify:
- \(A = 5\)
- \(B = 12\)
- For the first line \(C_1 = -20\)
- For the second line \(C_2 = 6\)

2. **Calculate the difference \(|C_2 - C_1|\)**:
\[ |C_2 - C_1| = |6 - (-20)| = |6 + 20| = |26| = 26 \]

3. **Calculate \(A^2 + B^2\)**:
\[ A^2 + B^2 = 5^2 + 12^2 = 25 + 144 = 169 \]

4. **Calculate \(\sqrt{A^2 + B^2}\)**:
\[ \sqrt{A^2 + B^2} = \sqrt{169} = 13 \]

5. **Substitute into the distance formula**:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} = \frac{26}{13} = 2 \]

6. **Conclusion**:
The distance between the two parallel lines is \(2\) units.