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

Options

A. `7:3`
B. `3:7`
C. `2:3`
D. None of these

Answer

To find the ratio in which the line \(3x + 4y + 2 = 0\) divides the distance between the lines \(3x + 4y + 5 = 0\) and \(3x + 4y - 5 = 0\), we will follow these steps:

### Step 1: Identify the equations of the lines
We have three lines:
1. Line 1: \(3x + 4y + 5 = 0\)
2. Line 2: \(3x + 4y - 5 = 0\)
3. Line 3: \(3x + 4y + 2 = 0\)

### Step 2: Find the distance between Line 1 and Line 2
The distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

For our lines:
- \(C_1 = 5\) (from Line 1)
- \(C_2 = -5\) (from Line 2)
- \(A = 3\), \(B = 4\)

Substituting these values into the distance formula:

\[ d = \frac{|-5 - 5|}{\sqrt{3^2 + 4^2}} = \frac{|-10|}{\sqrt{9 + 16}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2 \]

### Step 3: Find the distances from Line 3 to Line 1 and Line 2
Now, we need to find the distances from Line 3 to Line 1 and Line 3 to Line 2.

**Distance from Line 3 to Line 1:**
Using the same distance formula:

\[ d_1 = \frac{|C_1 - C_3|}{\sqrt{A^2 + B^2}} = \frac{|5 - (-2)|}{\sqrt{3^2 + 4^2}} = \frac{|5 + 2|}{\sqrt{25}} = \frac{7}{5} = 1.4 \]

**Distance from Line 3 to Line 2:**
\[ d_2 = \frac{|C_2 - C_3|}{\sqrt{A^2 + B^2}} = \frac{|-5 - (-2)|}{\sqrt{3^2 + 4^2}} = \frac{|-5 + 2|}{\sqrt{25}} = \frac{|-3|}{5} = \frac{3}{5} = 0.6 \]

### Step 4: Find the ratio
Now, we can find the ratio in which Line 3 divides the distance between Line 1 and Line 2:

\[ \text{Ratio} = \frac{d_1}{d_2} = \frac{1.4}{0.6} = \frac{14}{6} = \frac{7}{3} \]

### 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 \(7:3\).

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