Let `a_(r)`, `r=1,2,3,4` represent four positive real numbers other than unity such that `(log_(e)a_(r),(1)/(log_(e)a_(r)))` is a point on a circle whose abscissa of ends of diameter are `alpha` and `-alpha(alpha ne 0)` , then the minimum value of `a_(1)+a_(2)+a_(3)+a_(4)` is :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the minimum value of the sum \( a_1 + a_2 + a_3 + a_4 \) given that \( ( \log_e a_r, \frac{1}{\log_e a_r} ) \) lies on a circle with diameter endpoints at \( \alpha \) and \( -\alpha \).

### Step-by-Step Solution:

1. **Understanding the Circle**:
The points \( ( \alpha, 0 ) \) and \( ( -\alpha, 0 ) \) are the endpoints of the diameter of the circle. The center of the circle is at the origin \( (0, 0) \) and the radius is \( \alpha \).

2. **Equation of the Circle**:
The equation of the circle can be given as:
\[ x^2 + y^2 = \alpha^2 \]
where \( x = \log_e a_r \) and \( y = \frac{1}{\log_e a_r} \).

3. **Substituting the Point**:
Substituting \( x \) and \( y \) into the circle's equation:
\[ (\log_e a_r)^2 + \left(\frac{1}{\log_e a_r}\right)^2 = \alpha^2 \]

4. **Letting \( x = \log_e a_r \)**:
This gives us:
\[ x^2 + \frac{1}{x^2} = \alpha^2 \]
Multiplying through by \( x^2 \) (assuming \( x \neq 0 \)):
\[ x^4 - \alpha^2 x^2 + 1 = 0 \]

5. **Using Vieta's Formulas**:
Let \( x_1 = \log_e a_1, x_2 = \log_e a_2, x_3 = \log_e a_3, x_4 = \log_e a_4 \). The sum of the roots (from Vieta's formulas) of the polynomial \( x^4 - \alpha^2 x^2 + 1 = 0 \) is zero since there is no \( x^3 \) term.

Therefore:
\[ \log_e a_1 + \log_e a_2 + \log_e a_3 + \log_e a_4 = 0 \]

6. **Exponentiating**:
This implies:
\[ \log_e (a_1 a_2 a_3 a_4) = 0 \implies a_1 a_2 a_3 a_4 = 1 \]

7. **Applying AM-GM Inequality**:
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{a_1 + a_2 + a_3 + a_4}{4} \geq \sqrt[4]{a_1 a_2 a_3 a_4} \]
Since \( a_1 a_2 a_3 a_4 = 1 \):
\[ \frac{a_1 + a_2 + a_3 + a_4}{4} \geq \sqrt[4]{1} = 1 \]
Thus:
\[ a_1 + a_2 + a_3 + a_4 \geq 4 \]

8. **Minimum Value**:
The minimum value of \( a_1 + a_2 + a_3 + a_4 \) is \( 4 \), which occurs when \( a_1 = a_2 = a_3 = a_4 = 1 \). However, since the problem states that \( a_r \) are positive real numbers other than unity, we can approach values close to 1 to achieve this minimum.

### Conclusion:
The minimum value of \( a_1 + a_2 + a_3 + a_4 \) is:
\[ \boxed{4} \]