Statement-1 : IF `|z+1/z|` =a , where z is a complex number and a is a real number, the least and greatest values of |z| are ` (sqrt(a^(2)+4-a))/2 and (sqrt(a^(2)+ 4)+a)/2`
and Statement -2 : For a equal ot zero the greatest and the least values of |z| are equal .

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explanation for statement -8
B. Statement -1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement -8
C. Statement -1 is True, Statement -2 is False
D. Statement -1 is Flase, Statement -2 is True

Answer

To solve the problem, we need to analyze the given statements about the complex number \( z \) and its modulus.

### Step-by-Step Solution:

1. **Given Condition**: We start with the condition \( |z + \frac{1}{z}| = a \), where \( z \) is a complex number and \( a \) is a real number.

2. **Expressing \( z \)**: Let \( z = re^{i\theta} \), where \( r = |z| \) (the modulus of \( z \)) and \( \theta \) is the argument of \( z \). Then, we can express \( \frac{1}{z} = \frac{1}{re^{i\theta}} = \frac{1}{r} e^{-i\theta} \).

3. **Modulus Calculation**: Now we can write:
\[ |z + \frac{1}{z}| = |re^{i\theta} + \frac{1}{r} e^{-i\theta}| \]
This can be simplified as:
\[ |z + \frac{1}{z}| = |r + \frac{1}{r} e^{-2i\theta}| \]

4. **Using Triangle Inequality**: By applying the triangle inequality, we get:
\[ |z + \frac{1}{z}| \leq |z| + |\frac{1}{z}| = r + \frac{1}{r} \]
Thus, we have:
\[ a = |z + \frac{1}{z}| \leq r + \frac{1}{r} \]

5. **Finding Maximum and Minimum Values**: To find the least and greatest values of \( |z| \), we can set up the equation:
\[ r + \frac{1}{r} = a \]
Multiplying through by \( r \) gives:
\[ r^2 - ar + 1 = 0 \]
This is a quadratic equation in \( r \).

6. **Using the Quadratic Formula**: The roots of this quadratic equation can be found using the quadratic formula:
\[ r = \frac{a \pm \sqrt{a^2 - 4}}{2} \]
Therefore, the least and greatest values of \( |z| \) are:
\[ r_1 = \frac{a - \sqrt{a^2 - 4}}{2}, \quad r_2 = \frac{a + \sqrt{a^2 - 4}}{2} \]

7. **Verifying the Statements**:
- **Statement 1**: The least and greatest values of \( |z| \) are \( \frac{\sqrt{a^2 + 4 - a}}{2} \) and \( \frac{\sqrt{a^2 + 4 + a}}{2} \). This can be verified by substituting \( a \) into the expressions derived from the quadratic roots.
- **Statement 2**: For \( a = 0 \), the values become equal, confirming that the greatest and least values of \( |z| \) are indeed equal.

### Final Results:
- The least value of \( |z| \) is \( \frac{\sqrt{a^2 + 4 - a}}{2} \).
- The greatest value of \( |z| \) is \( \frac{\sqrt{a^2 + 4 + a}}{2} \).