Key Concepts and Formulas

• Properties of Complex Conjugates: For a complex number $z = a + bi$, its conjugate is $\overline{z} = a - bi$. Also, $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$, $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$, and $\operatorname{Re}(z) = \frac{z + \overline{z}}{2}$.
• Cartesian Form of a Complex Number: $z = x + iy$, where $x, y \in \mathbb{R}$, $x = \operatorname{Re}(z)$, and $y = \operatorname{Im}(z)$.
• Equation of a Circle: $(x - a)^2 + (y - b)^2 = r^2$, where $(a, b)$ is the center and $r$ is the radius, or the general form $x^2 + y^2 + 2gx + 2fy + c = 0$ with center $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$.

Step-by-Step Solution

1. Simplify the Given Equation Using Conjugate Properties

The given equation is: $\operatorname{Re}\left( \frac{z - 1}{2z + i} \right) + \operatorname{Re} \left( \frac{\overline{z} - 1}{2\overline{z} - i} \right) = 2$ Let $w = \frac{z - 1}{2z + i}$. Then $\overline{w} = \frac{\overline{z} - 1}{2\overline{z} - i}$. Why this step? Recognizing this relationship allows us to simplify the equation significantly.

The equation can be rewritten as: $\operatorname{Re}(w) + \operatorname{Re}(\overline{w}) = 2$ Since $\operatorname{Re}(w) = \operatorname{Re}(\overline{w})$, we have: $2 \operatorname{Re}(w) = 2$ $\operatorname{Re}(w) = 1$ Substitute $w$ back: $\operatorname{Re}\left( \frac{z - 1}{2z + i} \right) = 1$ Why this step? We've simplified the original equation into a more manageable form involving only one complex fraction.

2. Express $z$ in Cartesian Form

Substitute $z = x + iy$ into the equation: $\operatorname{Re}\left( \frac{(x + iy) - 1}{2(x + iy) + i} \right) = 1$ $\operatorname{Re}\left( \frac{(x - 1) + iy}{2x + i(2y + 1)} \right) = 1$ Why this step? This transformation allows us to work with real variables $x$ and $y$, making it easier to find the locus.

3. Calculate the Real Part of the Complex Fraction

Multiply the numerator and denominator by the conjugate of the denominator: $\operatorname{Re}\left( \frac{((x - 1) + iy)(2x - i(2y + 1))}{(2x + i(2y + 1))(2x - i(2y + 1))} \right) = 1$ Why this step? This is the standard method for finding the real part of a complex fraction.

The numerator becomes: $((x - 1) + iy)(2x - i(2y + 1)) = 2x(x - 1) + i(2xy) - i(x - 1)(2y + 1) + y(2y + 1)$ The real part of the numerator is $2x(x - 1) + y(2y + 1) = 2x^2 - 2x + 2y^2 + y$.

The denominator becomes: $(2x)^2 + (2y + 1)^2 = 4x^2 + 4y^2 + 4y + 1$

Now, substitute back into the equation: $\frac{2x^2 - 2x + 2y^2 + y}{4x^2 + 4y^2 + 4y + 1} = 1$

4. Formulate the Equation of the Locus

Multiply both sides by the denominator: $2x^2 - 2x + 2y^2 + y = 4x^2 + 4y^2 + 4y + 1$ Rearrange to get: $2x^2 + 2y^2 + 2x + 3y + 1 = 0$ Divide by 2: $x^2 + y^2 + x + \frac{3}{2}y + \frac{1}{2} = 0$ Why this step? We now have the equation in the standard form for a circle.

5. Determine the Center and Radius of the Circle

Comparing with $x^2 + y^2 + 2gx + 2fy + c = 0$: $2g = 1 \implies g = \frac{1}{2}$ $2f = \frac{3}{2} \implies f = \frac{3}{4}$ $c = \frac{1}{2}$

The center is $(a, b) = (-g, -f) = \left(-\frac{1}{2}, -\frac{3}{4}\right)$. The radius is $r = \sqrt{g^2 + f^2 - c} = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{3}{4}\right)^2 - \frac{1}{2}} = \sqrt{\frac{1}{4} + \frac{9}{16} - \frac{1}{2}} = \sqrt{\frac{4 + 9 - 8}{16}} = \sqrt{\frac{5}{16}} = \frac{\sqrt{5}}{4}$. Therefore, $r^2 = \frac{5}{16}$. Why this step? We extract the center and radius, which are needed for the final calculation.

6. Compute the Final Expression

We need to find $\frac{15ab}{r^2}$. $\frac{15ab}{r^2} = \frac{15 \left(-\frac{1}{2}\right) \left(-\frac{3}{4}\right)}{\frac{5}{16}} = \frac{15 \cdot \frac{3}{8}}{\frac{5}{16}} = \frac{45}{8} \cdot \frac{16}{5} = \frac{45 \cdot 2}{5} = 9 \cdot 2 = 18$ Why this step? This is the final calculation to obtain the answer.

Common Mistakes & Tips

• Be careful with signs when dealing with complex conjugates and Cartesian coordinate substitutions.
• Double-check algebraic manipulations, especially when expanding and simplifying expressions.
• Remember the formula for the radius and center of a circle given its general equation.

Summary

By using properties of complex conjugates, converting to Cartesian coordinates, and simplifying the resulting equation, we found that the locus is a circle with center $(-\frac{1}{2}, -\frac{3}{4})$ and radius squared $\frac{5}{16}$. Substituting these values into the expression $\frac{15ab}{r^2}$, we get 18.

The final answer is \boxed{18}, which corresponds to option (D).