The perpendicular distance between two parallel lines `3x + 4y -6=0` and `6x + 8y +7=0` is equal to

Options

A. `(19)/(10)` unit
B. `(19)/(2)` unit
C. `(19)/(5)` unit
D. `(10)/(19)` unit

Answer

To find the perpendicular distance between the two parallel lines given by the equations \(3x + 4y - 6 = 0\) and \(6x + 8y + 7 = 0\), we can follow these steps:

### Step 1: Confirm that the lines are parallel
The lines are in the form \(Ax + By + C = 0\). For the two lines to be parallel, the ratios of their coefficients \(A\) and \(B\) must be equal.

For the first line \(3x + 4y - 6 = 0\):
- \(A_1 = 3\)
- \(B_1 = 4\)

For the second line \(6x + 8y + 7 = 0\):
- \(A_2 = 6\)
- \(B_2 = 8\)

We check if \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\):
\[ \frac{3}{6} = \frac{1}{2} \quad \text{and} \quad \frac{4}{8} = \frac{1}{2} \]
Since both ratios are equal, the lines are parallel.

### Step 2: Identify the constants \(C_1\) and \(C_2\)
From the equations of the lines, we can express them in the form \(Ax + By + C = 0\).

For the first line:
\[ C_1 = -6 \]

For the second line:
\[ C_2 = 7 \]

### Step 3: Use the formula for the distance between two parallel lines
The formula for the distance \(D\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ D = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

### Step 4: Substitute the values into the formula
Here, \(A = 3\), \(B = 4\), \(C_1 = -6\), and \(C_2 = 7\).

Substituting these values into the formula:
\[ D = \frac{|7 - (-6)|}{\sqrt{3^2 + 4^2}} = \frac{|7 + 6|}{\sqrt{9 + 16}} = \frac{13}{\sqrt{25}} = \frac{13}{5} \]

### Step 5: Simplify the expression
The distance simplifies to:
\[ D = \frac{13}{5} = 2.6 \]

Thus, the perpendicular distance between the two parallel lines is \(2.6\).

### Final Answer:
The perpendicular distance between the two parallel lines is \(2.6\).

---