`a` and `b` are real numbers between 0 and 1 such that the points `Z_1 =a+ i`, `Z_2=1+ bi`, `Z_3= 0` form an equilateral triangle, then `a` and `b` are equal to

Options

A. `a=b=2+sqrt(3)`
B. `a=b=2-sqrt(3)`
C. `a=b=-2-sqrt(3)`
D. none of these

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the points \( Z_1 = a + i \), \( Z_2 = 1 + bi \), and \( Z_3 = 0 \) form an equilateral triangle.

### Step-by-Step Solution:

1. **Understanding the Points**:
- We have three complex numbers:
- \( Z_1 = a + i \)
- \( Z_2 = 1 + bi \)
- \( Z_3 = 0 \)

2. **Using the Property of Equilateral Triangles**:
- For three points \( Z_1, Z_2, Z_3 \) to form an equilateral triangle, the following condition must hold:
\[ |Z_1 - Z_2| = |Z_2 - Z_3| = |Z_3 - Z_1| \]

3. **Calculating the Distances**:
- Calculate \( |Z_1 - Z_2| \):
\[ Z_1 - Z_2 = (a - 1) + (1 - b)i \]
\[ |Z_1 - Z_2| = \sqrt{(a - 1)^2 + (1 - b)^2} \]

- Calculate \( |Z_2 - Z_3| \):
\[ Z_2 - Z_3 = 1 + bi \]
\[ |Z_2 - Z_3| = \sqrt{1^2 + b^2} = \sqrt{1 + b^2} \]

- Calculate \( |Z_3 - Z_1| \):
\[ Z_3 - Z_1 = - (a + i) \]
\[ |Z_3 - Z_1| = \sqrt{a^2 + 1} \]

4. **Setting the Distances Equal**:
- Set \( |Z_1 - Z_2| = |Z_2 - Z_3| \):
\[ \sqrt{(a - 1)^2 + (1 - b)^2} = \sqrt{1 + b^2} \]
- Squaring both sides:
\[ (a - 1)^2 + (1 - b)^2 = 1 + b^2 \]
\[ (a - 1)^2 + 1 - 2b + b^2 = 1 + b^2 \]
\[ (a - 1)^2 - 2b = 0 \]
\[ (a - 1)^2 = 2b \quad \text{(1)} \]

- Set \( |Z_2 - Z_3| = |Z_3 - Z_1| \):
\[ \sqrt{1 + b^2} = \sqrt{a^2 + 1} \]
- Squaring both sides:
\[ 1 + b^2 = a^2 + 1 \]
\[ b^2 = a^2 \quad \text{(2)} \]

5. **Solving the Equations**:
- From equation (2), we have \( b = a \) or \( b = -a \). Since \( a \) and \( b \) are both between 0 and 1, we take \( b = a \).
- Substitute \( b = a \) into equation (1):
\[ (a - 1)^2 = 2a \]
\[ a^2 - 2a + 1 = 2a \]
\[ a^2 - 4a + 1 = 0 \]

6. **Using the Quadratic Formula**:
- Solve for \( a \):
\[ a = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ a = \frac{4 \pm \sqrt{16 - 4}}{2} \]
\[ a = \frac{4 \pm \sqrt{12}}{2} \]
\[ a = \frac{4 \pm 2\sqrt{3}}{2} \]
\[ a = 2 \pm \sqrt{3} \]

7. **Finding Valid Values**:
- Since \( a \) must be between 0 and 1, we take:
\[ a = 2 - \sqrt{3} \]
- Since \( b = a \), we also have:
\[ b = 2 - \sqrt{3} \]

### Final Answer:
Thus, the values of \( a \) and \( b \) are:
\[ a = b = 2 - \sqrt{3} \]