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

Explanation

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 can follow these steps:

### Step 1: Identify 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: Confirm the lines are parallel
All three lines are in the form \(Ax + By + C = 0\) where \(A = 3\) and \(B = 4\). Since the coefficients of \(x\) and \(y\) are the same, the lines are parallel.

### Step 3: Calculate the distance between Line 1 and Line 2
The formula for the distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:

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

For Line 1 and Line 2:
- \(C_1 = 5\)
- \(C_2 = -5\)

Thus, the distance \(D_{12}\) is:

\[ D_{12} = \frac{|-5 - 5|}{\sqrt{3^2 + 4^2}} = \frac{|-10|}{\sqrt{9 + 16}} = \frac{10}{5} = 2 \]

### Step 4: Calculate the distance between Line 2 and Line 3
Now, we calculate the distance between Line 2 and Line 3 using the same formula:

For Line 2 and Line 3:
- \(C_1 = -5\)
- \(C_2 = 2\)

Thus, the distance \(D_{23}\) is:

\[ D_{23} = \frac{|2 - (-5)|}{\sqrt{3^2 + 4^2}} = \frac{|2 + 5|}{\sqrt{9 + 16}} = \frac{7}{5} \]

### Step 5: Find the ratio \(p:q\)
Let \(p\) be the distance from Line 1 to Line 3, and \(q\) be the distance from Line 3 to Line 2. We have:

- \(p = D_{12} = 2\)
- \(q = D_{23} = \frac{7}{5}\)

Now, we can express the ratio \(p:q\):

\[ \frac{p}{q} = \frac{2}{\frac{7}{5}} = 2 \cdot \frac{5}{7} = \frac{10}{7} \]

Thus, the ratio \(p:q\) is:

\[ p:q = 10:7 \]

### 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 \(10:7\).

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