The distance between the lines `y=mx+c_1` and `y=mx+c_2` is

Options

A. `(c_1-c_2)/(sqrt(m^(2)+1))`
B. `|(c_1-c_2)|/(sqrt(1+m^(2)))`
C. `(c_2-c_1)/(sqrt(1+m^(2)))`
D. 0

Explanation

To find the distance between the two parallel lines given by the equations \( y = mx + c_1 \) and \( y = mx + c_2 \), we can follow these steps:

### Step 1: Identify the equations of the lines
The two lines are:
1. \( y = mx + c_1 \)
2. \( y = mx + c_2 \)

### Step 2: Recognize that the lines are parallel
Since both lines have the same slope \( m \), they are parallel. The distance between two parallel lines can be calculated using a specific formula.

### Step 3: Convert the equations to standard form
We can rewrite the equations in the standard form \( Ax + By + C = 0 \):
1. For the first line:
\[ mx - y + c_1 = 0 \quad \Rightarrow \quad A_1 = m, B_1 = -1, C_1 = c_1 \]
2. For the second line:
\[ mx - y + c_2 = 0 \quad \Rightarrow \quad A_2 = m, B_2 = -1, C_2 = c_2 \]

### Step 4: 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 5: Substitute the values into the formula
In our case:
- \( C_1 = c_1 \)
- \( C_2 = c_2 \)
- \( A = m \)
- \( B = -1 \)

Thus, the distance \( d \) becomes:
\[ d = \frac{|c_2 - c_1|}{\sqrt{m^2 + (-1)^2}} = \frac{|c_2 - c_1|}{\sqrt{m^2 + 1}} \]

### Final Result
The distance between the lines \( y = mx + c_1 \) and \( y = mx + c_2 \) is:
\[ d = \frac{|c_2 - c_1|}{\sqrt{m^2 + 1}} \]

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