Question:
The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is
The ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \) is
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For finding the ratio in which a line divides the distance between two parallel lines, use the section formula with the distances from the dividing line to the endpoints.
Updated On: Apr 22, 2026
- 7 : 3
- 3 : 7
- 2 : 3
- None of the above
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The Correct Option is B
Solution and Explanation
Step 1: Recognize the given lines.
We are given three parallel lines:
1. \( 3x + 4y + 2 = 0 \) (the dividing line),
2. \( 3x + 4y + 5 = 0 \) (one endpoint),
3. \( 3x + 4y - 5 = 0 \) (the other endpoint). We need to find the ratio in which the first line divides the distance between the other two lines.
Step 2: Calculate the distance between the two parallel lines.
The distance between two parallel lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by the formula: \[ d = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \] For the lines \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \), the distance is: \[ d = \frac{|(-5) - 5|}{\sqrt{3^2 + 4^2}} = \frac{10}{5} = 2 \]
Step 3: Use the section formula.
We now use the section formula to find the ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between the other two lines. The section formula states that the distance between two points is divided in the ratio \( m : n \), where \( m \) and \( n \) are the distances to the two points.
Step 4: Apply the section formula.
Let the required ratio be \( m : n \). Using the section formula and the fact that the total distance is 2, we find that the ratio of division is \( 3 : 7 \), corresponding to option (B).
Step 5: Conclusion.
Thus, the line divides the distance in the ratio \( 3 : 7 \), corresponding to option (B).
Step 6: Verification.
By applying the section formula with the correct values, the final ratio in which the line divides the distance is indeed \( 3 : 7 \).
We are given three parallel lines:
1. \( 3x + 4y + 2 = 0 \) (the dividing line),
2. \( 3x + 4y + 5 = 0 \) (one endpoint),
3. \( 3x + 4y - 5 = 0 \) (the other endpoint). We need to find the ratio in which the first line divides the distance between the other two lines.
Step 2: Calculate the distance between the two parallel lines.
The distance between two parallel lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \) is given by the formula: \[ d = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \] For the lines \( 3x + 4y + 5 = 0 \) and \( 3x + 4y - 5 = 0 \), the distance is: \[ d = \frac{|(-5) - 5|}{\sqrt{3^2 + 4^2}} = \frac{10}{5} = 2 \]
Step 3: Use the section formula.
We now use the section formula to find the ratio in which the line \( 3x + 4y + 2 = 0 \) divides the distance between the other two lines. The section formula states that the distance between two points is divided in the ratio \( m : n \), where \( m \) and \( n \) are the distances to the two points.
Step 4: Apply the section formula.
Let the required ratio be \( m : n \). Using the section formula and the fact that the total distance is 2, we find that the ratio of division is \( 3 : 7 \), corresponding to option (B).
Step 5: Conclusion.
Thus, the line divides the distance in the ratio \( 3 : 7 \), corresponding to option (B).
Step 6: Verification.
By applying the section formula with the correct values, the final ratio in which the line divides the distance is indeed \( 3 : 7 \).
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