Two complex numbers A and B, are where k is a constant. If `AB-15=60`, what is the value of k? (Note: `i^(2)=-1`)

Answer

To solve the problem, we need to find the value of \( k \) given the complex numbers \( A = 3 - 4i \) and \( B = 9 + ki \). We know from the problem statement that \( AB - 15 = 60 \), which means \( AB = 75 \).

### Step-by-Step Solution:

1. **Set up the equation:**
\[ AB = 75 \]

2. **Multiply the complex numbers \( A \) and \( B \):**
\[ A = 3 - 4i \quad \text{and} \quad B = 9 + ki \]
\[ AB = (3 - 4i)(9 + ki) \]

3. **Use the distributive property to expand the multiplication:**
\[ AB = 3 \cdot 9 + 3 \cdot ki - 4i \cdot 9 - 4i \cdot ki \]
\[ = 27 + 3ki - 36i - 4k(i^2) \]

4. **Substitute \( i^2 = -1 \):**
\[ -4k(i^2) = -4k(-1) = 4k \]
Thus, we can rewrite \( AB \):
\[ AB = 27 + 3ki - 36i + 4k \]
\[ = (27 + 4k) + (3k - 36)i \]

5. **Set the real and imaginary parts equal to the corresponding parts of 75:**
Since \( AB = 75 \) can be considered as \( 75 + 0i \), we can equate the real and imaginary parts:
- Real part:
\[ 27 + 4k = 75 \]
- Imaginary part:
\[ 3k - 36 = 0 \]

6. **Solve the real part equation:**
\[ 4k = 75 - 27 \]
\[ 4k = 48 \]
\[ k = \frac{48}{4} = 12 \]

7. **Solve the imaginary part equation:**
\[ 3k - 36 = 0 \]
\[ 3k = 36 \]
\[ k = \frac{36}{3} = 12 \]

Both methods yield the same result.

### Final Answer:
\[ k = 12 \]