The distance between the lines `5x+12y+65=0` and `5x+12y-39 = 0` is :

Options

A. 4
B. 16
C. 2
D. 8

Answer

To find the distance between the two parallel lines given by the equations \(5x + 12y + 65 = 0\) and \(5x + 12y - 39 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The equations of the lines can be written in the standard form:
- Line 1: \(5x + 12y + 65 = 0\)
- Line 2: \(5x + 12y - 39 = 0\)

From these equations, we can identify:
- \(a_1 = 5\), \(b_1 = 12\), \(c_1 = 65\)
- \(a_2 = 5\), \(b_2 = 12\), \(c_2 = -39\)

### Step 2: Verify that the lines are parallel
Since the coefficients of \(x\) and \(y\) are the same in both equations, the lines are parallel.

### Step 3: Use the formula for the distance between two parallel lines
The formula for the distance \(d\) between two parallel lines given by \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) is:

\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]

### Step 4: Substitute the values into the formula
Substituting the values we identified:
- \(c_1 = 65\)
- \(c_2 = -39\)

We calculate:

\[ d = \frac{|65 - (-39)|}{\sqrt{5^2 + 12^2}} \]

### Step 5: Simplify the expression
Calculating the numerator:

\[ |65 + 39| = |104| = 104 \]

Calculating the denominator:

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

### Step 6: Final calculation of distance
Now substituting back into the distance formula:

\[ d = \frac{104}{13} = 8 \]

### Conclusion
The distance between the two parallel lines is \(8\).

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