Distance between the lines `3x+4y-5=0` and `6x+8y-45=0` is

Options

A. `1/3`
B. `7/2`
C. 1
D. `2/7`

Answer

To find the distance between the two lines given by the equations \(3x + 4y - 5 = 0\) and \(6x + 8y - 45 = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are:
1. Line 1: \(3x + 4y - 5 = 0\)
2. Line 2: \(6x + 8y - 45 = 0\)

### Step 2: Check if the lines are parallel
To check if the lines are parallel, we can compare their slopes. The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope.

For Line 1:
\[ 4y = -3x + 5 \implies y = -\frac{3}{4}x + \frac{5}{4} \]
Thus, the slope \(m_1 = -\frac{3}{4}\).

For Line 2:
\[ 8y = -6x + 45 \implies y = -\frac{6}{8}x + \frac{45}{8} \implies y = -\frac{3}{4}x + \frac{45}{8} \]
Thus, the slope \(m_2 = -\frac{3}{4}\).

Since \(m_1 = m_2\), 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 of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

### Step 4: Rewrite the equations in the standard form
From the first line, we have:
- \(C_1 = -5\)

From the second line, we can rewrite it as:
\[ 6x + 8y - 45 = 0 \implies 3x + 4y - \frac{45}{2} = 0 \]
Thus, \(C_2 = -\frac{45}{2}\).

### Step 5: Identify \(A\), \(B\), \(C_1\), and \(C_2\)
From the equations, we have:
- \(A = 3\)
- \(B = 4\)
- \(C_1 = -5\)
- \(C_2 = -\frac{45}{2}\)

### Step 6: Calculate the distance
Now, we can substitute these values into the distance formula:
\[ d = \frac{|-\frac{45}{2} - (-5)|}{\sqrt{3^2 + 4^2}} = \frac{|\frac{-45 + 10}{2}|}{\sqrt{9 + 16}} = \frac{|\frac{-35}{2}|}{\sqrt{25}} = \frac{\frac{35}{2}}{5} = \frac{35}{10} = \frac{7}{2} \]

### Final Answer
The distance between the lines is \(\frac{7}{2}\).

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