Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = a + bi$, the modulus is $|z| = \sqrt{a^2 + b^2}$, and $|z|^2 = z\bar{z}$.
• Equation of a Circle: The equation $|z - z_0| = r$ represents a circle in the complex plane centered at $z_0$ with radius $r$. Thus, $|z - z_0|^2 = r^2$.
• Properties of Conjugates: $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$, $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$, $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$, and $\overline{\bar{z}} = z$.

Step-by-Step Solution

Step 1: Express the first condition using the modulus definition.

The complex number $\alpha$ lies on the circle $|z - z_0|^2 = 4$. Substituting $z = \alpha$ and $z_0 = 1+i$, we have $|\alpha - (1+i)|^2 = 4.$ Using the property $|w|^2 = w\bar{w}$ with $w = \alpha - (1+i)$, we can write this as $(\alpha - (1+i))(\overline{\alpha - (1+i)}) = 4.$ Applying the conjugate properties, we get $(\alpha - (1+i))(\bar{\alpha} - \overline{(1+i)}) = 4.$ Since $\overline{1+i} = 1-i$, $(\alpha - (1+i))(\bar{\alpha} - (1-i)) = 4.$ Expanding the product gives $\alpha\bar{\alpha} - \alpha(1-i) - (1+i)\bar{\alpha} + (1+i)(1-i) = 4.$ We know that $\alpha\bar{\alpha} = |\alpha|^2$ and $(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2$. Thus, $|\alpha|^2 - \alpha(1-i) - \bar{\alpha}(1+i) + 2 = 4.$ Rearranging the terms, we have $|\alpha|^2 - [\alpha(1-i) + \bar{\alpha}(1+i)] = 2. \quad \text{(1)}$ Reasoning: We are rewriting the given geometric condition using the modulus definition and conjugate properties to obtain an algebraic equation involving $|\alpha|^2$, $\alpha$, and $\bar{\alpha}$. This transformation helps us proceed towards solving for $|\alpha|^2$.

Step 2: Express the second condition using the modulus definition.

The complex number $\frac{1}{\bar{\alpha}}$ lies on the circle $|z - z_0|^2 = 16$. Substituting $z = \frac{1}{\bar{\alpha}}$ and $z_0 = 1+i$, we have $\left|\frac{1}{\bar{\alpha}} - (1+i)\right|^2 = 16.$ Using the property $|w|^2 = w\bar{w}$ with $w = \frac{1}{\bar{\alpha}} - (1+i)$, we can write this as $\left(\frac{1}{\bar{\alpha}} - (1+i)\right)\left(\overline{\frac{1}{\bar{\alpha}} - (1+i)}\right) = 16.$ Applying the conjugate properties, we get $\left(\frac{1}{\bar{\alpha}} - (1+i)\right)\left(\frac{1}{\alpha} - (1-i)\right) = 16.$ Multiplying both sides by $\alpha\bar{\alpha} = |\alpha|^2$ yields $\left(\frac{1 - (1+i)\bar{\alpha}}{\bar{\alpha}}\right)\left(\frac{1 - (1-i)\alpha}{\alpha}\right) = 16.$ $\frac{(1 - (1+i)\bar{\alpha})(1 - (1-i)\alpha)}{\alpha\bar{\alpha}} = 16.$ $(1 - (1+i)\bar{\alpha})(1 - (1-i)\alpha) = 16|\alpha|^2.$ Expanding the product gives $1 - (1-i)\alpha - (1+i)\bar{\alpha} + (1+i)(1-i)\alpha\bar{\alpha} = 16|\alpha|^2.$ We know $(1+i)(1-i) = 2$ and $\alpha\bar{\alpha} = |\alpha|^2$. Therefore, $1 - (1-i)\alpha - (1+i)\bar{\alpha} + 2|\alpha|^2 = 16|\alpha|^2.$ $1 - [\alpha(1-i) + \bar{\alpha}(1+i)] = 14|\alpha|^2. \quad \text{(2)}$ Reasoning: Similar to Step 1, we rewrite the second geometric condition into an algebraic equation. The main difference is the presence of $\frac{1}{\bar{\alpha}}$, which requires careful application of conjugate properties.

Step 3: Solve the system of equations.

We have two equations: $|\alpha|^2 - [\alpha(1-i) + \bar{\alpha}(1+i)] = 2 \quad \text{(1)}$ $1 - [\alpha(1-i) + \bar{\alpha}(1+i)] = 14|\alpha|^2 \quad \text{(2)}$ From equation (1), we can write $[\alpha(1-i) + \bar{\alpha}(1+i)] = |\alpha|^2 - 2.$ Substituting this into equation (2), we get $1 - (|\alpha|^2 - 2) = 14|\alpha|^2.$ $1 - |\alpha|^2 + 2 = 14|\alpha|^2.$ $3 = 15|\alpha|^2.$ $|\alpha|^2 = \frac{3}{15} = \frac{1}{5}.$ Reasoning: By isolating the common term $[\alpha(1-i) + \bar{\alpha}(1+i)]$, we can substitute and eliminate it, thereby reducing the system to a single equation with one unknown, $|\alpha|^2$.

Step 4: Calculate $100|\alpha|^2$.

We are asked to find the value of $100|\alpha|^2$. Since $|\alpha|^2 = \frac{1}{5}$, we have $100|\alpha|^2 = 100 \times \frac{1}{5} = 20.$

Common Mistakes & Tips

• Conjugate Properties: Be extremely careful when applying conjugate properties, especially with fractions and sums/differences.
• Algebraic Expansion: Double-check your algebraic expansions to avoid sign errors or missed terms.
• Modulus Calculation: Ensure you correctly calculate $|z_0|^2$ and $|\alpha|^2$ using the appropriate formulas.

Summary

We transformed the given geometric conditions into algebraic equations by using the properties of modulus and conjugates of complex numbers. By solving the resulting system of equations, we determined the value of $|\alpha|^2$ and subsequently calculated $100|\alpha|^2$.

The final answer is \boxed{20}.