Key Concepts and Formulas

• Modulus-Conjugate Property: For any complex number $w$, $|w|^2 = w\overline{w}$. This allows us to eliminate the modulus and work with algebraic expressions.
• Properties of Conjugates:
  • $\overline{w_1 \pm w_2} = \overline{w_1} \pm \overline{w_2}$
  • $\overline{w^n} = (\overline{w})^n$
• Complex Number Representation: A complex number $z$ can be represented as $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part. Its conjugate is $\overline{z} = x - iy$. Therefore, $z + \overline{z} = 2x = 2\text{Re}(z)$.

Step-by-Step Solution

Step 1: Start with the Given Equation

We are given the equation: $|z^2 - 1| = |z|^2 + 1$

Step 2: Square Both Sides

Why? Squaring eliminates the modulus, allowing us to use the property $|w|^2 = w\overline{w}$. $|z^2 - 1|^2 = (|z|^2 + 1)^2$

Step 3: Apply the Modulus-Conjugate Property

Why? We use the property $|w|^2 = w\overline{w}$ on the left-hand side. $(z^2 - 1)(\overline{z^2 - 1}) = (|z|^2 + 1)^2$ Using conjugate properties, $\overline{z^2 - 1} = \overline{z^2} - \overline{1} = (\overline{z})^2 - 1$. $(z^2 - 1)((\overline{z})^2 - 1) = (|z|^2 + 1)^2$

Step 4: Expand Both Sides

Why? Expanding allows us to simplify and combine terms. Expanding the left side: $z^2(\overline{z})^2 - z^2 - (\overline{z})^2 + 1$ Since $z\overline{z} = |z|^2$, we have $z^2(\overline{z})^2 = (z\overline{z})^2 = |z|^4$. So, the left side becomes: $|z|^4 - z^2 - (\overline{z})^2 + 1$

Expanding the right side: $(|z|^2 + 1)^2 = |z|^4 + 2|z|^2 + 1$

Now, equate the expanded left and right sides: $|z|^4 - z^2 - (\overline{z})^2 + 1 = |z|^4 + 2|z|^2 + 1$

Step 5: Simplify the Equation

Why? Canceling identical terms simplifies the equation. Subtract $|z|^4$ and $1$ from both sides: $-z^2 - (\overline{z})^2 = 2|z|^2$ Multiply by -1: $z^2 + (\overline{z})^2 = -2|z|^2$

Step 6: Substitute $|z|^2 = z\overline{z}$ and Rearrange

Why? Expressing the equation in terms of $z$ and $\overline{z}$ helps in simplification. $z^2 + (\overline{z})^2 = -2z\overline{z}$ $z^2 + 2z\overline{z} + (\overline{z})^2 = 0$

Step 7: Factor the Expression

Why? Factoring reveals a crucial relationship between $z$ and $\overline{z}$. $(z + \overline{z})^2 = 0$ $z + \overline{z} = 0$

Step 8: Interpret the Result

Why? We analyze the equation to determine the locus of $z$. Let $z = x + iy$, then $\overline{z} = x - iy$. Substituting into $z + \overline{z} = 0$: $(x + iy) + (x - iy) = 0$ $2x = 0$ $x = 0$ This means the real part of $z$ is zero, so $z$ is purely imaginary. Therefore, $z$ lies on the imaginary axis.

Common Mistakes & Tips

• Remember the conjugate: $|w|^2 = w\overline{w}$, not $w^2$.
• Conjugate properties: Ensure you correctly apply the properties of conjugates, especially when dealing with sums and powers.
• Algebraic Manipulation: Double-check signs and terms during expansion and simplification to avoid errors.

Summary

By squaring the given equation, applying the modulus-conjugate property, and simplifying, we arrived at the condition $z + \overline{z} = 0$. This implies that the real part of $z$ is zero, meaning $z$ is a purely imaginary number. Therefore, $z$ lies on the imaginary axis. This corresponds to option (B).

The final answer is \boxed{the imaginary axis}.