Find the perpendicular distance between the lines
`y=mx+c, y=mx+d`

Answer

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

### Step 1: Rewrite the equations in standard form
The equations of the lines can be rewritten in the standard form \( Ax + By + C = 0 \).

For the first line \( y = mx + c \):
\[ mx - y + c = 0 \]
This gives us \( A_1 = m, B_1 = -1, C_1 = c \).

For the second line \( y = mx + d \):
\[ mx - y + d = 0 \]
This gives us \( A_2 = m, B_2 = -1, C_2 = d \).

### Step 2: Identify the coefficients
From the standard forms, we have:
- \( A = m \)
- \( B = -1 \)
- \( C_1 = c \)
- \( C_2 = d \)

### Step 3: 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:
\[ D = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ D = \frac{|c - d|}{\sqrt{m^2 + (-1)^2}} \]
This simplifies to:
\[ D = \frac{|c - d|}{\sqrt{m^2 + 1}} \]

### Final Result
Thus, the perpendicular distance between the lines \( y = mx + c \) and \( y = mx + d \) is:
\[ D = \frac{|c - d|}{\sqrt{m^2 + 1}} \]

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