If the distance between the parallel lines `3x+4y+7=0` and `ax+y+b=0` is 1, the intergral value of b is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the integral value of \( b \) such that the distance between the parallel lines \( 3x + 4y + 7 = 0 \) and \( ax + y + b = 0 \) is equal to 1.

### Step-by-Step Solution:

1. **Identify the Coefficients:**
The first line is given by the equation:
\[ 3x + 4y + 7 = 0 \]
Here, the coefficients are \( A_1 = 3 \), \( B_1 = 4 \), and \( C_1 = 7 \).

2. **Form of the Second Line:**
The second line is given as:
\[ ax + y + b = 0 \]
For the lines to be parallel, the ratios of the coefficients of \( x \) and \( y \) must be equal:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \]
This implies:
\[ \frac{3}{a} = \frac{4}{1} \implies 3 = 4a \implies a = \frac{3}{4} \]

3. **Substituting \( a \) in the Second Line:**
Now substituting \( a \) into the second line's equation:
\[ \frac{3}{4}x + y + b = 0 \implies 3x + 4y + 4b = 0 \]
Thus, the coefficients for the second line are \( A_2 = 3 \), \( B_2 = 4 \), and \( C_2 = 4b \).

4. **Distance Formula Between Parallel Lines:**
The distance \( d \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]
Substituting the values:
\[ d = \frac{|7 - 4b|}{\sqrt{3^2 + 4^2}} = \frac{|7 - 4b|}{\sqrt{9 + 16}} = \frac{|7 - 4b|}{5} \]

5. **Setting the Distance Equal to 1:**
We know that the distance is equal to 1:
\[ \frac{|7 - 4b|}{5} = 1 \]
Multiplying both sides by 5:
\[ |7 - 4b| = 5 \]

6. **Solving the Absolute Value Equation:**
This gives us two cases to solve:
- Case 1: \( 7 - 4b = 5 \)
- Case 2: \( 7 - 4b = -5 \)

**Case 1:**
\[ 7 - 4b = 5 \implies -4b = 5 - 7 \implies -4b = -2 \implies b = \frac{1}{2} \]

**Case 2:**
\[ 7 - 4b = -5 \implies -4b = -5 - 7 \implies -4b = -12 \implies b = 3 \]

7. **Finding Integral Values:**
The integral values of \( b \) from both cases are \( \frac{1}{2} \) and \( 3 \). Since we are looking for integral values, we take \( b = 3 \).

### Final Answer:
The integral value of \( b \) is \( \boxed{3} \).