Equation of line mid-parallel to the lines `3x+4y=12` and `3x+4y=2` is

Options

A. `4x+3y=7`
B. `3x+4y+7=0`
C. `3x+4y=7`
D. `3x+4y=0`

Answer

To find the equation of the line that is mid-parallel to the lines \(3x + 4y = 12\) and \(3x + 4y = 2\), we can follow these steps:

### Step 1: Identify the equations of the parallel lines
The given equations are:
1. \(3x + 4y = 12\)
2. \(3x + 4y = 2\)

### Step 2: Rewrite the equations in standard form
We can rewrite these equations in the form \(Ax + By + C = 0\):
1. \(3x + 4y - 12 = 0\) (Let \(C_1 = -12\))
2. \(3x + 4y - 2 = 0\) (Let \(C_2 = -2\))

### Step 3: Calculate the distance between the two parallel lines
The formula for the distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]
In our case:
- \(A = 3\)
- \(B = 4\)
- \(C_1 = -12\)
- \(C_2 = -2\)

Substituting these values into the formula:
\[ d = \frac{|-12 - (-2)|}{\sqrt{3^2 + 4^2}} = \frac{|-12 + 2|}{\sqrt{9 + 16}} = \frac{|-10|}{\sqrt{25}} = \frac{10}{5} = 2 \]

### Step 4: Find the mid-distance
Since we want the mid-parallel line, we take half of the distance:
\[ \text{Mid distance} = \frac{d}{2} = \frac{2}{2} = 1 \]

### Step 5: Set up the equation for the mid-parallel line
The equation of the mid-parallel line can be expressed as:
\[ 3x + 4y + C = 0 \]
We need to find the value of \(C\). The distance from this mid-parallel line to either of the original lines should be 1.

### Step 6: Use the distance formula again
Using the distance formula again, we set the distance to 1:
\[ 1 = \frac{|C_1 - C|}{\sqrt{A^2 + B^2}} \quad \text{(using } C_1 = -12\text{)} \]
Substituting the values:
\[ 1 = \frac{|-12 - C|}{5} \]
This gives us two equations to solve:
1. \(-12 - C = 5\) (1)
2. \(-12 - C = -5\) (2)

### Step 7: Solve the equations
**From equation (1):**
\[ -12 - C = 5 \implies -C = 5 + 12 \implies -C = 17 \implies C = -17 \]

**From equation (2):**
\[ -12 - C = -5 \implies -C = -5 + 12 \implies -C = 7 \implies C = -7 \]

### Step 8: Choose the correct value of C
The mid-parallel line should be between the two original lines. Therefore, we take \(C = -7\).

### Final Equation
Thus, the equation of the mid-parallel line is:
\[ 3x + 4y - 7 = 0 \]

### Summary
The equation of the line mid-parallel to the lines \(3x + 4y = 12\) and \(3x + 4y = 2\) is:
\[ \boxed{3x + 4y - 7 = 0} \]