Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is $|z| = \sqrt{x^2 + y^2}$. Geometrically, this represents the distance of the point $(x, y)$ from the origin.
• Apollonius's Circle: The equation $\left| \frac{z - z_1}{z - z_2} \right| = k$, where $k > 0$ and $k \neq 1$, represents a circle.
• Equation of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.

Step-by-Step Solution

1. Rewrite the Given Equation The given equation is: $\left| {{{z - 2} \over {z - 3}}} \right| = 2$ We rewrite it as: $|z - 2| = 2 |z - 3|$

• Explanation: We use the property that $\left| \frac{w_1}{w_2} \right| = \frac{|w_1|}{|w_2|}$ and multiply both sides by $|z - 3|$ to obtain an equation relating distances.

2. Substitute $z = x + iy$ Let $z = x + iy$, where $x$ and $y$ are real numbers. Substituting this into the equation, we get: $|(x + iy) - 2| = 2 |(x + iy) - 3|$ $|(x - 2) + iy| = 2 |(x - 3) + iy|$

• Explanation: We substitute $z$ with its Cartesian form to convert the complex equation into a form we can manipulate algebraically using real numbers.

3. Apply the Modulus Definition and Square Using the definition of the modulus, $|a + bi| = \sqrt{a^2 + b^2}$, we get: $\sqrt{(x - 2)^2 + y^2} = 2 \sqrt{(x - 3)^2 + y^2}$ Squaring both sides: $(x - 2)^2 + y^2 = 4[(x - 3)^2 + y^2]$

• Explanation: Squaring both sides eliminates the square roots and makes the equation easier to simplify.

4. Expand and Simplify Expanding the terms, we get: $x^2 - 4x + 4 + y^2 = 4(x^2 - 6x + 9 + y^2)$ $x^2 - 4x + 4 + y^2 = 4x^2 - 24x + 36 + 4y^2$ Rearranging the terms: $3x^2 + 3y^2 - 20x + 32 = 0$

• Explanation: We expand the squared terms and rearrange the equation into a general form for a conic section.

5. Convert to Standard Circle Form Divide by 3 to make the coefficients of $x^2$ and $y^2$ equal to 1: $x^2 + y^2 - \frac{20}{3}x + \frac{32}{3} = 0$

• Explanation: We divide by 3 to obtain the standard general form of a circle equation.

6. Find the Center and Radius Comparing with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = -\frac{20}{3}$, $2f = 0$, and $c = \frac{32}{3}$. Thus, $g = -\frac{10}{3}$, $f = 0$, and $c = \frac{32}{3}$.

The center is $(\alpha, \beta) = (-g, -f) = \left(\frac{10}{3}, 0\right)$.

The radius is $\gamma = \sqrt{g^2 + f^2 - c} = \sqrt{\left(-\frac{10}{3}\right)^2 + 0^2 - \frac{32}{3}} = \sqrt{\frac{100}{9} - \frac{32}{3}} = \sqrt{\frac{100 - 96}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3}$.

• Explanation: We use the coefficients of the standard equation to calculate the center and radius of the circle.

7. Calculate $3(\alpha + \beta + \gamma)$ We have $\alpha = \frac{10}{3}$, $\beta = 0$, and $\gamma = \frac{2}{3}$. Therefore, $3(\alpha + \beta + \gamma) = 3\left(\frac{10}{3} + 0 + \frac{2}{3}\right) = 3\left(\frac{12}{3}\right) = 3(4) = 12$

• Explanation: We substitute the calculated values of the center and radius into the expression we want to find.

Common Mistakes & Tips:

• Squaring: When squaring both sides of the equation, make sure to square the entire expression, including any constants.
• Algebraic Manipulation: Be careful with algebraic manipulations, especially when expanding and simplifying the equation.
• Signs: Pay close attention to signs when finding the center from the general form of the circle equation. The center is $(-g, -f)$.

Summary

By converting the complex equation into its Cartesian equivalent and simplifying, we identified it as a circle with center $(\frac{10}{3}, 0)$ and radius $\frac{2}{3}$. Therefore, $3(\alpha + \beta + \gamma) = 12$.

The final answer is \boxed{12}, which corresponds to option (A).