Question

The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between the lines \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is

• A. 3 : 7
• B. 2 : 3
• C. 1 : 2
• D. 2 : 5

Hint:

To solve problems involving the distance between parallel lines, use the formula for the distance between two parallel lines and then use the ratio of the distances.

Answer 1

The Correct Option is A

The given equation of the lines are: 1. \( 3x + 4y + 5 = 0 \) 2. \( 3x + 4y - 5 = 0 \) 3. \( 3x + 4y + 2 = 0 \) We need to find the ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between the two parallel lines. Step 1: Formula for distance between two parallel lines The formula for the distance between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by: \[ \text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \] For the lines \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \), we have: \[ A = 3, \ B = 4, \ C_1 = 5, \ C_2 = -5 \] Thus, the distance between the two lines is: \[ \text{Distance} = \frac{|(-5) - 5|}{\sqrt{3^2 + 4^2}} = \frac{| -10 |}{5} = \frac{10}{5} = 2 \] Step 2: Ratio in which the line divides the distance The equation of the line dividing the distance is \( 3x + 4y + 2 = 0 \). The distance from this line to \( 3x + 4y + 5 = 0 \) is: \[ \text{Distance} = \frac{|2 - 5|}{5} = \frac{3}{5} \] And the distance from this line to \( 3x + 4y - 5 = 0 \) is: \[ \text{Distance} = \frac{|2 - (-5)|}{5} = \frac{7}{5} \] Now, the required ratio of the distances is: \[ \frac{\frac{3}{5}}{\frac{7}{5}} = \frac{3}{7} \] Thus, the ratio in which the line divides the distance is \( 3 : 7 \).