Find the perpendicular distance between the lines `3x+4y+9=0` and to `6x+8y+15=0` is

Options

A. `3//2`
B. `3//10`
C. 6
D. none of these

Answer

To find the perpendicular distance between the lines \(3x + 4y + 9 = 0\) and \(6x + 8y + 15 = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are:
1. \(3x + 4y + 9 = 0\) (let's call this Line 1)
2. \(6x + 8y + 15 = 0\) (let's call this Line 2)

### Step 2: Rewrite the equations in the standard form
We can rewrite the equations in the form \(Ax + By + C = 0\):
- Line 1: \(3x + 4y + 9 = 0\) can be rewritten as \(3x + 4y = -9\).
- Line 2: \(6x + 8y + 15 = 0\) can be rewritten as \(6x + 8y = -15\).

### Step 3: Check if the lines are parallel
To check if the lines are parallel, we compare their slopes. The slope of a line in the form \(Ax + By + C = 0\) is given by \(-\frac{A}{B}\).
- For Line 1: Slope = \(-\frac{3}{4}\)
- For Line 2: Slope = \(-\frac{6}{8} = -\frac{3}{4}\)

Since both slopes are equal, the lines are parallel.

### Step 4: Use the formula for the distance between two parallel lines
The formula for the perpendicular 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 5: Identify the coefficients and constants
From the equations:
- For Line 1: \(C_1 = -9\)
- For Line 2: \(C_2 = -15\)
- Here, \(A = 3\) and \(B = 4\).

### Step 6: Substitute the values into the distance formula
Substituting the values into the formula:
\[ d = \frac{|-15 - (-9)|}{\sqrt{3^2 + 4^2}} = \frac{|-15 + 9|}{\sqrt{9 + 16}} = \frac{|-6|}{\sqrt{25}} = \frac{6}{5} \]

### Step 7: Final result
Thus, the perpendicular distance between the two lines is:
\[ d = \frac{6}{5} \]