Let relation `R` defined on set of natural number is given by `R = {(a,b) : a^(2)+b^(2)-4bsqrt(3)-2a sqrt(3)+15=0}` then relation `R` is

Options

A. only symmetric
B. only reflexive
C. only transitive
D. Symmetric and transitive

Explanation

To solve the problem, we need to analyze the relation \( R \) defined by the equation:

\[ a^2 + b^2 - 4b\sqrt{3} - 2a\sqrt{3} + 15 = 0 \]

### Step 1: Rearranging the Equation

We start by rearranging the equation to complete the square for both \( a \) and \( b \).

1. For \( a \):
\[ a^2 - 2a\sqrt{3} \quad \text{(we will complete the square)} \]
To complete the square, we add and subtract \( (\sqrt{3})^2 = 3 \):
\[ a^2 - 2a\sqrt{3} + 3 - 3 \]
This can be rewritten as:
\[ (a - \sqrt{3})^2 - 3 \]

2. For \( b \):
\[ b^2 - 4b\sqrt{3} \quad \text{(we will complete the square)} \]
To complete the square, we add and subtract \( (2\sqrt{3})^2 = 12 \):
\[ b^2 - 4b\sqrt{3} + 12 - 12 \]
This can be rewritten as:
\[ (b - 2\sqrt{3})^2 - 12 \]

### Step 2: Substitute Back into the Equation

Now, substituting these completed squares back into the original equation:

\[ ((a - \sqrt{3})^2 - 3) + ((b - 2\sqrt{3})^2 - 12) + 15 = 0 \]

This simplifies to:

\[ (a - \sqrt{3})^2 + (b - 2\sqrt{3})^2 = 0 \]

### Step 3: Analyzing the Equation

The equation \( (a - \sqrt{3})^2 + (b - 2\sqrt{3})^2 = 0 \) implies that both squares must equal zero:

1. \( (a - \sqrt{3})^2 = 0 \) implies \( a = \sqrt{3} \)
2. \( (b - 2\sqrt{3})^2 = 0 \) implies \( b = 2\sqrt{3} \)

### Step 4: Checking the Values of \( a \) and \( b \)

Since \( a \) and \( b \) must be natural numbers, we need to check if \( \sqrt{3} \) and \( 2\sqrt{3} \) are natural numbers:

- \( \sqrt{3} \) is not a natural number.
- \( 2\sqrt{3} \) is also not a natural number.

### Conclusion

Since there are no pairs \( (a, b) \) in the set of natural numbers that satisfy the equation, the relation \( R \) is empty.

Thus, the relation \( R \) is:

- **Empty Relation**