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 lines \( y = mx + c_1 \) and \( y = mx + c_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the equations of the lines**:
We have two lines given by the equations:
\[ y = mx + c_1 \quad \text{(Line 1)} \]
\[ y = mx + c_2 \quad \text{(Line 2)} \]

2. **Recognize that the lines are parallel**:
Since both lines have the same slope \( m \), they are parallel.

3. **Convert the equations to standard form**:
The standard form of a line is given by \( Ax + By + C = 0 \). We can rewrite the equations as:
\[ -mx + y - c_1 = 0 \quad \text{(Line 1 in standard form)} \]
\[ -mx + y - c_2 = 0 \quad \text{(Line 2 in standard form)} \]

4. **Identify coefficients for the distance formula**:
From the standard form \( Ax + By + C = 0 \), we can identify:
- For Line 1: \( A = -m \), \( B = 1 \), \( C = -c_1 \)
- For Line 2: \( A = -m \), \( B = 1 \), \( C = -c_2 \)

5. **Use the formula for the distance between two parallel lines**:
The distance \( d \) between two parallel lines given by \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
Here, \( C_1 = -c_1 \) and \( C_2 = -c_2 \).

6. **Substitute the values into the distance formula**:
\[ d = \frac{|-c_2 - (-c_1)|}{\sqrt{(-m)^2 + 1^2}} = \frac{|c_1 - c_2|}{\sqrt{m^2 + 1}} \]

7. **Final result**:
Thus, the distance between the two lines is given by:
\[ d = \frac{|c_1 - c_2|}{\sqrt{1 + m^2}} \]