If the terms of G.P. `sqrt(a-x),sqrt(x),sqrt(a+x)`.. are all in integers where `agto` then the least composite odd integral value of `a` is

Options

A. `9`
B. `15`
C. `45`
D. `49`

Explanation

To solve the problem, we need to determine the least composite odd integral value of \( a \) such that the terms \( \sqrt{a - x}, \sqrt{x}, \sqrt{a + x} \) are all integers and \( a > 0 \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
The terms \( \sqrt{a - x}, \sqrt{x}, \sqrt{a + x} \) need to be in Arithmetic Progression (AP). For three terms \( A, B, C \) to be in AP, the condition is:
\[ 2B = A + C \]
Here, let \( A = \sqrt{a - x} \), \( B = \sqrt{x} \), and \( C = \sqrt{a + x} \).

2. **Setting Up the Equation**:
From the AP condition, we have:
\[ 2\sqrt{x} = \sqrt{a - x} + \sqrt{a + x} \]

3. **Squaring Both Sides**:
To eliminate the square roots, we square both sides:
\[ 4x = (\sqrt{a - x} + \sqrt{a + x})^2 \]
Expanding the right-hand side:
\[ 4x = (a - x) + (a + x) + 2\sqrt{(a - x)(a + x)} \]
Simplifying gives:
\[ 4x = 2a + 2\sqrt{(a - x)(a + x)} \]

4. **Rearranging the Equation**:
Rearranging the equation, we get:
\[ 2\sqrt{(a - x)(a + x)} = 4x - 2a \]
Dividing by 2:
\[ \sqrt{(a - x)(a + x)} = 2x - a \]

5. **Squaring Again**:
Squaring both sides again gives:
\[ (a - x)(a + x) = (2x - a)^2 \]
Expanding both sides:
\[ a^2 - x^2 = 4x^2 - 4ax + a^2 \]
Simplifying leads to:
\[ -x^2 = 4x^2 - 4ax \]
Rearranging gives:
\[ 5x^2 - 4ax = 0 \]

6. **Factoring**:
Factoring out \( x \):
\[ x(5x - 4a) = 0 \]
This gives us two cases: \( x = 0 \) or \( 5x = 4a \).

7. **Finding \( x \)**:
From \( 5x = 4a \), we have:
\[ x = \frac{4a}{5} \]

8. **Substituting Back**:
Now substituting \( x = \frac{4a}{5} \) back into the expressions for \( \sqrt{a - x}, \sqrt{x}, \sqrt{a + x} \):
- \( \sqrt{x} = \sqrt{\frac{4a}{5}} \)
- \( \sqrt{a - x} = \sqrt{a - \frac{4a}{5}} = \sqrt{\frac{a}{5}} \)
- \( \sqrt{a + x} = \sqrt{a + \frac{4a}{5}} = \sqrt{\frac{9a}{5}} \)

9. **Ensuring Integer Values**:
For these square roots to be integers, \( a \) must be of the form \( a = 5n^2 \) where \( n \) is a natural number.

10. **Finding Composite Odd Values**:
- For \( n = 1 \): \( a = 5 \) (not composite)
- For \( n = 2 \): \( a = 20 \) (composite but not odd)
- For \( n = 3 \): \( a = 45 \) (composite and odd)

Thus, the least composite odd integral value of \( a \) is:
\[ \boxed{45} \]