Prove that distance between two parallel lines `ax+by+c_1=0` and `ax+by+c_2=0` is given by `|c_1-c_2|/(sqrt(a^2+b^2))`

Options

A. `|(c_(1)+c_(2))/(sqrt(a^(2)+b^(2)))|`
B. `|(c_(1)-c_(2))/(sqrt(a^(2)+b^(2)))|`
C. `|(c_(1)-c_(2))/(sqrt(a^(2)-b^(2)))|`
D. `|(c_(1)-c_(2))/(sqrt(a^(2)+b^(2)))|`

Answer

To prove that the distance \( d \) between two parallel lines given by the equations \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by the formula:

\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]

we can follow these steps:

### Step 1: Understand the equations of the lines
The two lines are given as:
1. Line 1: \( ax + by + c_1 = 0 \)
2. Line 2: \( ax + by + c_2 = 0 \)

Since both lines have the same coefficients for \( x \) and \( y \) (i.e., \( a \) and \( b \)), they are parallel.

### Step 2: Find the perpendicular distance from a point to a line
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \) is given by:

\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

### Step 3: Choose a point on one of the lines
To find the distance between the two parallel lines, we can choose a point on one of the lines. Let's choose a point on Line 1, \( ax + by + c_1 = 0 \). We can set \( x = 0 \) to find a point on this line:

\[ b y + c_1 = 0 \implies y = -\frac{c_1}{b} \quad \text{(assuming \( b \neq 0 \))} \]

Thus, the point on Line 1 is \( (0, -\frac{c_1}{b}) \).

### Step 4: Calculate the distance from this point to Line 2
Now, we will calculate the distance from the point \( (0, -\frac{c_1}{b}) \) to Line 2, \( ax + by + c_2 = 0 \):

Using the distance formula:

\[ d = \frac{|a(0) + b\left(-\frac{c_1}{b}\right) + c_2|}{\sqrt{a^2 + b^2}} \]

This simplifies to:

\[ d = \frac{|-c_1 + c_2|}{\sqrt{a^2 + b^2}} = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \]

### Step 5: Finalize the result
Since \( |c_2 - c_1| = |c_1 - c_2| \), we can write:

\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]

Thus, we have proved that the distance between the two parallel lines is given by:

\[ d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \]