Find the distance between two parallel lines 5x + 12y -30 =0 and 5x+12y - 4=0

Options

A. 3
B. 7
C. `5/2`
D. 2

Answer

To find the distance between the two parallel lines given by the equations \(5x + 12y - 30 = 0\) and \(5x + 12y - 4 = 0\), we can use the formula for the distance \(D\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\):

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

### Step-by-Step Solution:

1. **Identify the coefficients**:
From the equations of the lines:
- For the first line \(5x + 12y - 30 = 0\), we have:
- \(A = 5\)
- \(B = 12\)
- \(C_1 = -30\)
- For the second line \(5x + 12y - 4 = 0\), we have:
- \(C_2 = -4\)

2. **Calculate the difference of \(C_1\) and \(C_2\)**:
\[ |C_1 - C_2| = |-30 - (-4)| = |-30 + 4| = |-26| = 26 \]

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

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

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

### Final Answer:
The distance between the two parallel lines is \(2\) units.

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