If `alpha, beta " are the roots of the equation " lx^(2) - mx + m =0, l!=m,l!=0,` then which one of the following statement is correct ?

Options

A. `sqrt(alpha/beta) + sqrt(beta/alpha)-sqrt(m/l)=0`
B. ` sqrt(alpha/beta) + sqrt(beta/alpha)+sqrt(m/l)=0`
C. `sqrt(alpha +beta)/(alpha beta)-sqrt(m/l)= 0 `
D. The arithmetic mean of `alpha and beta` is the same as their geometric mean

Explanation

To solve the problem, we need to analyze the quadratic equation given and utilize the properties of its roots.

### Step-by-Step Solution:

1. **Identify the given quadratic equation**:
The equation is given as:
\[ lx^2 - mx + m = 0 \]
where \( l \neq m \) and \( l \neq 0 \).

2. **Use Vieta's formulas**:
According to Vieta's formulas, for a quadratic equation of the form \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

For our equation:
- \( \alpha + \beta = \frac{m}{l} \)
- \( \alpha \beta = \frac{m}{l} \)

3. **Express \( \sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} \)**:
We can express this as:
\[ \sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} = \frac{\sqrt{\alpha^2} + \sqrt{\beta^2}}{\sqrt{\alpha \beta}} = \frac{\alpha + \beta}{\sqrt{\alpha \beta}} \]

4. **Substituting the values from Vieta's formulas**:
Now substituting the values we found:
\[ \sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} = \frac{\frac{m}{l}}{\sqrt{\frac{m}{l}}} \]

5. **Simplifying the expression**:
This simplifies to:
\[ \sqrt{\frac{m}{l}} \]

6. **Final equation**:
We can write:
\[ \sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} - \sqrt{\frac{m}{l}} = 0 \]

### Conclusion:
From the above steps, we can conclude that the correct statement among the options provided is:
\[ \sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} = \sqrt{\frac{m}{l}} \]