The domain of R is:
`R={x,y):x+2y=8 , x,y in N}`

Options

A. {2,4,8}
B. {2,4,6,8}
C. {2,4,6}
D. {1,2,3,4}

Answer

To find the domain \( R \) defined by the equation \( x + 2y = 8 \) where \( x \) and \( y \) are natural numbers, we will follow these steps:

### Step 1: Understand the equation
We start with the equation given in the problem:
\[ x + 2y = 8 \]
This equation relates \( x \) and \( y \).

### Step 2: Express \( y \) in terms of \( x \)
Rearranging the equation to solve for \( y \):
\[ 2y = 8 - x \]
\[ y = \frac{8 - x}{2} \]

### Step 3: Determine the values of \( x \)
Since \( x \) and \( y \) must be natural numbers (i.e., positive integers), we need to find values of \( x \) such that \( y \) remains a natural number.

### Step 4: Find possible values of \( x \)
We will substitute natural numbers for \( x \) and check if \( y \) is also a natural number:
- If \( x = 1 \):
\[ y = \frac{8 - 1}{2} = \frac{7}{2} = 3.5 \quad (\text{not a natural number}) \]
- If \( x = 2 \):
\[ y = \frac{8 - 2}{2} = \frac{6}{2} = 3 \quad (\text{natural number}) \]
- If \( x = 3 \):
\[ y = \frac{8 - 3}{2} = \frac{5}{2} = 2.5 \quad (\text{not a natural number}) \]
- If \( x = 4 \):
\[ y = \frac{8 - 4}{2} = \frac{4}{2} = 2 \quad (\text{natural number}) \]
- If \( x = 5 \):
\[ y = \frac{8 - 5}{2} = \frac{3}{2} = 1.5 \quad (\text{not a natural number}) \]
- If \( x = 6 \):
\[ y = \frac{8 - 6}{2} = \frac{2}{2} = 1 \quad (\text{natural number}) \]
- If \( x = 7 \):
\[ y = \frac{8 - 7}{2} = \frac{1}{2} = 0.5 \quad (\text{not a natural number}) \]
- If \( x = 8 \):
\[ y = \frac{8 - 8}{2} = \frac{0}{2} = 0 \quad (\text{not a natural number}) \]

### Step 5: Collect valid pairs
From the calculations above, the valid pairs \((x, y)\) where both \( x \) and \( y \) are natural numbers are:
- \( (2, 3) \)
- \( (4, 2) \)
- \( (6, 1) \)

### Step 6: Identify the domain
The values of \( x \) that correspond to valid pairs are:
- \( x = 2 \)
- \( x = 4 \)
- \( x = 6 \)

Thus, the domain \( R \) is:
\[ R = \{ (2, 3), (4, 2), (6, 1) \} \]
The values of \( x \) in the domain are \( 2, 4, 6 \).

### Final Answer
The domain of \( R \) is:
\[ \{2, 4, 6\} \]