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 \(5x + 12y - 20 = 0\) and \(5x + 12y + 6 = 0\), we can follow these steps:

### Step 1: Identify the coefficients and constant terms
The given lines are:
1. \(l_1: 5x + 12y - 20 = 0\)
2. \(l_2: 5x + 12y + 6 = 0\)

From these equations, we can identify:
- Coefficient \(a = 5\)
- Coefficient \(b = 12\)
- Constant term of the first line \(c_1 = -20\)
- Constant term of the second line \(c_2 = 6\)

### Step 2: Use the formula for the distance between two parallel lines
The formula for the distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:

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

### Step 3: Substitute the values into the formula
Here, we substitute \(C_1 = -20\) and \(C_2 = 6\):

\[ d = \frac{|-20 - 6|}{\sqrt{5^2 + 12^2}} \]

### Step 4: Calculate the numerator
Calculate the absolute difference in the constant terms:

\[ |-20 - 6| = |-26| = 26 \]

### Step 5: Calculate the denominator
Now, calculate \(A^2 + B^2\):

\[ \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

### Step 6: Calculate the distance
Now, substitute the values back into the distance formula:

\[ d = \frac{26}{13} = 2 \]

### Final Answer
The distance between the parallel lines \(5x + 12y - 20 = 0\) and \(5x + 12y + 6 = 0\) is \(2\).

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