Find the distance between the two parallel straight lines y = mx + c and y = mx + d ? `["Assume" c gt d]`

Options

A. `(((c-d))/((1+m^(2))^(1//2)))`
B. `(((d-c))/((1+m^(2))^(1//2)))`
C. `((d)/((1+m^(2))^(1//2)))`
D. `((-d)/((1+m^(2))^(1//2)))`

Answer

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

### Step 1: Understand the equations of the lines
The two lines are:
1. \( y = mx + c \)
2. \( y = mx + d \)

Since both lines have the same slope \( m \), they are parallel.

### Step 2: Identify the distance formula for parallel lines
The distance \( P \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) can be calculated using the formula:
\[ P = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

### Step 3: Rewrite the equations in the required form
We can rewrite the equations of the lines in the standard form:
1. \( mx - y + c = 0 \) (which gives \( C_1 = c \))
2. \( mx - y + d = 0 \) (which gives \( C_2 = d \))

### Step 4: Apply the distance formula
Now, we can apply the distance formula:
- Here, \( A = m \), \( B = -1 \), \( C_1 = c \), and \( C_2 = d \).
- Thus, the distance \( P \) is given by:
\[ P = \frac{|d - c|}{\sqrt{m^2 + (-1)^2}} = \frac{|d - c|}{\sqrt{m^2 + 1}} \]

### Step 5: Simplify the expression
Since we are given that \( c > d \), we can replace \( |c - d| \) with \( c - d \):
\[ P = \frac{c - d}{\sqrt{m^2 + 1}} \]

### Final Result
Therefore, the distance between the two parallel lines \( y = mx + c \) and \( y = mx + d \) is:
\[ P = \frac{c - d}{\sqrt{1 + m^2}} \]