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`

Options

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

Answer

To solve the problem, we need to find the ratio in which the line \(3x + 4y + 2 = 0\) divides the distance between the lines \(3x + 4y + 5 = 0\) and \(3x + 4y - 5 = 0\).

### Step-by-step Solution:

1. **Identify the Lines**:
- We have three lines:
- Line 1: \(3x + 4y + 5 = 0\) (let's call this Line A)
- Line 2: \(3x + 4y - 5 = 0\) (let's call this Line B)
- Line 3: \(3x + 4y + 2 = 0\) (let's call this Line C)

2. **Check Parallelism**:
- All three lines are parallel because they have the same coefficients for \(x\) and \(y\) (i.e., \(3\) and \(4\)). The only difference is in the constant term.

3. **Calculate the Distance Between Line A and Line B**:
- The formula for the distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
- For Line A and Line B:
- \(C_1 = 5\) (from Line A)
- \(C_2 = -5\) (from Line B)
- \(A = 3\), \(B = 4\)
- Therefore, the distance \(d\) is:
\[ d = \frac{|(-5) - 5|}{\sqrt{3^2 + 4^2}} = \frac{|-10|}{\sqrt{9 + 16}} = \frac{10}{5} = 2 \]

4. **Calculate the Distances from Line C to Lines A and B**:
- **Distance from Line C to Line A**:
- Using the same formula:
- \(C_1 = 5\) (from Line A)
- \(C_2 = -2\) (from Line C)
- Therefore, the distance \(d_1\) is:
\[ d_1 = \frac{|(-2) - 5|}{\sqrt{3^2 + 4^2}} = \frac{|-7|}{\sqrt{9 + 16}} = \frac{7}{5} \]

- **Distance from Line C to Line B**:
- \(C_1 = -5\) (from Line B)
- \(C_2 = -2\) (from Line C)
- Therefore, the distance \(d_2\) is:
\[ d_2 = \frac{|(-2) - (-5)|}{\sqrt{3^2 + 4^2}} = \frac{|3|}{\sqrt{9 + 16}} = \frac{3}{5} \]

5. **Find the Ratio**:
- The ratio in which Line C divides the distance between Line A and Line B is given by the distances \(d_1\) and \(d_2\):
\[ \text{Ratio} = \frac{d_2}{d_1} = \frac{\frac{3}{5}}{\frac{7}{5}} = \frac{3}{7} \]

### Final Answer:
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 \(3:7\).