Key Concepts and Formulas

• Argument of a Complex Number: For a complex number $z = x + iy$, $\arg(z) = \tan^{-1}\left(\frac{y}{x}\right)$.
• Argument of Quotient: $\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$.
• Locus of a Circle: The equation $|z - z_0| = r$ represents a circle with center $z_0$ and radius $r$. In Cartesian coordinates, $(x-a)^2 + (y-b)^2 = r^2$ represents a circle with center $(a,b)$ and radius $r$.
• Minimum Distance to a Circle: The minimum distance from a point $P$ to a circle with center $C$ and radius $r$ is $|PC| - r$ if $P$ is outside the circle and $0$ if $P$ is inside the circle.

Step-by-Step Solution

Step 1: Express $z$ in Cartesian form and substitute into the given equation. Let $z = x + iy$. We are given $\arg\left(\frac{z-2}{z+2}\right) = \frac{\pi}{4}$. We substitute $z = x + iy$ to get: $\arg\left(\frac{(x+iy)-2}{(x+iy)+2}\right) = \frac{\pi}{4}$ $\arg\left(\frac{(x-2)+iy}{(x+2)+iy}\right) = \frac{\pi}{4}$ Why: This allows us to manipulate the equation algebraically by converting from complex form to real and imaginary components.

Step 2: Apply the argument property for division. Using the property $\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$, we get: $\arg((x-2)+iy) - \arg((x+2)+iy) = \frac{\pi}{4}$ Why: This simplifies the equation by separating the complex fraction into two individual arguments.

Step 3: Convert arguments to $\tan^{-1}$ form. Using the definition $\arg(a+ib) = \tan^{-1}\left(\frac{b}{a}\right)$, we have: $\tan^{-1}\left(\frac{y}{x-2}\right) - \tan^{-1}\left(\frac{y}{x+2}\right) = \frac{\pi}{4}$ Why: This converts the equation into a form that can be further simplified using trigonometric identities.

Step 4: Use the trigonometric identity for $\tan^{-1}A - \tan^{-1}B$. Using the identity $\tan^{-1}A - \tan^{-1}B = \tan^{-1}\left(\frac{A-B}{1+AB}\right)$, we get: $\tan^{-1}\left(\frac{\frac{y}{x-2} - \frac{y}{x+2}}{1 + \frac{y}{x-2} \cdot \frac{y}{x+2}}\right) = \frac{\pi}{4}$ Why: This combines the two $\tan^{-1}$ terms into one, simplifying the equation.

Step 5: Simplify the expression algebraically. Taking the tangent of both sides: $\frac{\frac{y(x+2) - y(x-2)}{(x-2)(x+2)}}{1 + \frac{y^2}{(x-2)(x+2)}} = \tan\left(\frac{\pi}{4}\right) = 1$ $\frac{y(x+2) - y(x-2)}{(x-2)(x+2) + y^2} = 1$ $\frac{yx + 2y - yx + 2y}{x^2 - 4 + y^2} = 1$ $\frac{4y}{x^2 + y^2 - 4} = 1$ $4y = x^2 + y^2 - 4$ $x^2 + y^2 - 4y - 4 = 0$ Why: This series of algebraic manipulations simplifies the equation to a more recognizable form.

Step 6: Complete the square to find the equation of the circle. $x^2 + (y^2 - 4y + 4) = 4 + 4$ $x^2 + (y - 2)^2 = 8$ Why: This puts the equation in standard circle form, making it easy to identify the center and radius.

Step 7: Identify the circle's center and radius. The equation $x^2 + (y - 2)^2 = 8$ represents a circle with center $(0, 2)$ and radius $r = \sqrt{8} = 2\sqrt{2}$. Why: This gives us the geometric parameters needed to solve the minimization problem.

Step 8: Find the distance between the circle's center and the given point. We want to minimize $|z - (9\sqrt{2} + 2i)|^2$, which is the squared distance between a point $z$ on the circle and the point $P_0 = (9\sqrt{2}, 2)$. The distance between the center $C = (0, 2)$ and $P_0$ is: $CP_0 = \sqrt{(9\sqrt{2} - 0)^2 + (2 - 2)^2} = \sqrt{(9\sqrt{2})^2} = 9\sqrt{2}$ Why: We need this distance to determine the minimum distance from the point to the circle.

Step 9: Calculate the minimum distance from the point to the circle. Since $CP_0 = 9\sqrt{2} > 2\sqrt{2} = r$, the point $P_0$ is outside the circle. The minimum distance is $CP_0 - r = 9\sqrt{2} - 2\sqrt{2} = 7\sqrt{2}$. Why: Applying the formula for minimum distance from a point to a circle.

Step 10: Square the minimum distance. The problem asks for the square of the minimum distance, which is $(7\sqrt{2})^2 = 49 \cdot 2 = 98$. Why: This gives us the final answer to the problem.

Step 11: Verify that the solution satisfies the condition y>0. The closest point on the circle to $(9\sqrt{2}, 2)$ is $(2\sqrt{2}, 2)$. Since the y coordinate is 2, y>0. Why: The solution must satisfy the original conditions.

Common Mistakes & Tips

• Remember to check if the point you are minimizing to is inside or outside the circle.
• Be careful with algebraic manipulations, especially when dealing with fractions and square roots.
• Always check that the final solution makes sense geometrically.

Summary

We found the locus of $z$ to be a circle with center $(0, 2)$ and radius $2\sqrt{2}$. We then found the distance between the center of the circle and the point $(9\sqrt{2}, 2)$, and used this to find the minimum distance from the point to the circle. Squaring this minimum distance gives us the final answer of 98.

The final answer is $\boxed{98}$.