If z 1 ​ =a+ib and z 2 ​ =c+id are complex numbers such that ∣z 1 ​ ∣=∣z 2 ​ ∣=1 and Re(z 1 ​ z 2 ​ ​ )=0, then the pair of complex numbers w 1 ​ =a+ic and w 2 ​ =b+id satisfy

Options

A. ∣w
1
​
∣=1
B. ∣w
2
​
∣=1
C. Re(w
1
​

w
2
​

​
)=0
D. All the above

Answer

To solve the problem, we start with the given complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \), where \( |z_1| = |z_2| = 1 \) and \( \text{Re}(z_1 \overline{z_2}) = 0 \). We need to analyze the implications of these conditions and find out what the pair of complex numbers \( w_1 = a + ic \) and \( w_2 = b + id \) satisfy.

### Step-by-Step Solution:

1. **Understanding the Magnitudes**:
Since \( |z_1| = 1 \) and \( |z_2| = 1 \), we have:
\[ |z_1|^2 = a^2 + b^2 = 1 \]
\[ |z_2|^2 = c^2 + d^2 = 1 \]

2. **Real Part Condition**:
The condition \( \text{Re}(z_1 \overline{z_2}) = 0 \) implies:
\[ z_1 \overline{z_2} = (a + ib)(c - id) = ac + bd + i(bc - ad) \]
The real part is \( ac + bd \), and since this equals zero, we have:
\[ ac + bd = 0 \quad \text{(1)} \]

3. **Expressing \( w_1 \) and \( w_2 \)**:
We have:
\[ w_1 = a + ic \]
\[ w_2 = b + id \]

4. **Finding the Product \( w_1 \overline{w_2} \)**:
To find \( w_1 \overline{w_2} \):
\[ \overline{w_2} = b - id \]
Thus,
\[ w_1 \overline{w_2} = (a + ic)(b - id) = ab - aid + ibc + c(d) \]
This simplifies to:
\[ w_1 \overline{w_2} = ab + cd + i(bc - ad) \]

5. **Finding the Real Part**:
The real part of \( w_1 \overline{w_2} \) is:
\[ \text{Re}(w_1 \overline{w_2}) = ab + cd \quad \text{(2)} \]

6. **Using the Condition from Step 2**:
From equation (1), we know \( ac + bd = 0 \). We can rearrange this to find:
\[ ac = -bd \]

7. **Relating \( ab + cd \) to Zero**:
Since \( a^2 + b^2 = 1 \) and \( c^2 + d^2 = 1 \), we can express \( cd \) in terms of \( ac \):
\[ ab + cd = ab - ac = ab - (-bd) = ab + bd \]
But since \( ac + bd = 0 \), we can conclude:
\[ ab + cd = 0 \]

8. **Final Conclusion**:
Thus, we find that:
\[ \text{Re}(w_1 \overline{w_2}) = 0 \]

### Summary:
The pair of complex numbers \( w_1 = a + ic \) and \( w_2 = b + id \) satisfy:
\[ \text{Re}(w_1 \overline{w_2}) = 0 \]