Question

The distance between the parallel lines \[ 3x - 4y + 7 = 0 \quad {and} \quad 3x - 4y + 5 = 0 { is } \frac{a}{b}. { Value of } a + b { is:} \]

• A. 2
• B. 5
• C. 7
• D. 3

Hint:

To find the distance between two parallel lines, use the formula \( d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \).

Answer 1

The Correct Option is C

Step 1: 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 2: For the given lines, \( A = 3 \), \( B = -4 \), \( C_1 = 7 \), and \( C_2 = 5 \). Substituting these values into the formula: \[ d = \frac{|7 - 5|}{\sqrt{3^2 + (-4)^2}} = \frac{2}{\sqrt{9 + 16}} = \frac{2}{5}. \] Thus, \( a = 2 \) and \( b = 5 \), so \( a + b = 7 \).