Question

If \( Z = 1 + i \) and \( Z - 24\overline{Z} = \lambda Z^2 \), find \( \lambda \).

Hint:

When solving complex number equations, separate the real and imaginary parts to solve for the unknowns.

Answer 1

Step 1: Express \( Z \) and \( \overline{Z} \).
We are given that \( Z = 1 + i \). The conjugate of \( Z \), denoted \( \overline{Z} \), is: \[ \overline{Z} = 1 - i \]
Step 2: Substitute \( Z \) and \( \overline{Z} \) into the equation.
We are also given the equation \( Z - 24\overline{Z} = \lambda Z^2 \). Substituting the values of \( Z \) and \( \overline{Z} \): \[ (1 + i) - 24(1 - i) = \lambda (1 + i)^2 \]
Step 3: Simplify the equation.
First, simplify the left-hand side: \[ (1 + i) - 24(1 - i) = 1 + i - 24 + 24i = -23 + 25i \] Now, simplify the right-hand side: \[ \lambda (1 + i)^2 = \lambda (1 + 2i - 1) = \lambda (2i) \] Thus, the equation becomes: \[ -23 + 25i = \lambda \cdot 2i \]
Step 4: Solve for \( \lambda \).
Equating the real and imaginary parts, we get: \[ \text{Real part: } -23 = 0 \quad \text{(This is already satisfied.)} \] \[ \text{Imaginary part: } 25 = 2\lambda \quad \Rightarrow \quad \lambda = \frac{25}{2} \]