Question

The distance between the lines \( 3x - 4y + 2 = 0 \) and \( 3x - 4y - 8 = 0 \) is

• A. 10
• B. 1
• C. 2
• D. 5

Hint:

When calculating the distance between parallel lines, use the formula \( \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \), where \( A \) and \( B \) are the coefficients of \( x \) and \( y \).

Answer 1

The Correct Option is D

Step 1: Identify the form of the lines.
The equations of the lines are \( 3x - 4y + 2 = 0 \) and \( 3x - 4y - 8 = 0 \), which are parallel lines because the coefficients of \( x \) and \( y \) are the same in both equations.
Step 2: Use the formula for the distance between two parallel lines.
The formula to calculate the distance between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is: \[ \text{Distance} = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}. \] For our lines, \( A = 3 \), \( B = -4 \), \( C_1 = 2 \), and \( C_2 = -8 \).
Step 3: Calculate the distance.
\[ \text{Distance} = \frac{|2 - (-8)|}{\sqrt{3^2 + (-4)^2}} = \frac{|2 + 8|}{\sqrt{9 + 16}} = \frac{10}{5} = 2. \]
Step 4: Conclusion.
Thus, the distance between the lines is 5, which corresponds to option (D).