Key Concepts and Formulas

• Circle Equation: The equation $|z - z_0| = r$ represents a circle in the complex plane with center $z_0$ and radius $r$. If $z = x + iy$ and $z_0 = x_0 + iy_0$, then the Cartesian form is $(x - x_0)^2 + (y - y_0)^2 = r^2$.
• Real Part of a Complex Number: For $z = x + iy$, the real part is given by Re$(z) = x$. The inequality Re$(z) \ge a$ represents the region to the right of and including the vertical line $x = a$.
• Imaginary Part of a Complex Number: For $z = x + iy$, the imaginary part is Im$(z) = y$. Also, $z - \overline{z} = 2i \text{Im}(z) = 2iy$. The inequality $|z - \overline{z}| \ge a$ is equivalent to $|2y| \ge a$ or $|y| \ge a/2$, representing two horizontal regions.

Step-by-Step Solution

Let $z = x + iy$, where $x$ and $y$ are real numbers.

Step 1: Analyzing Set $S_1$

The set $S_1$ is defined as $S_1 = \{ z \in C||z - 3 - 2i{|^2} = 8\}$.

• Explanation: We need to interpret this equation geometrically. The expression $|z - (3 + 2i)|^2 = 8$ represents the set of all points $z$ whose distance from the complex number $3 + 2i$ is $\sqrt{8} = 2\sqrt{2}$. This is a circle in the complex plane.
• Applying the formula: Taking the square root, we have $|z - (3 + 2i)| = 2\sqrt{2}$. This is a circle with center $3 + 2i$ and radius $2\sqrt{2}$.
• Geometrical Interpretation: The circle has center $(3, 2)$ and radius $r = 2\sqrt{2}$.
• Cartesian Equation: Substituting $z = x + iy$ into the equation $|z - (3 + 2i)|^2 = 8$, we get $|(x + iy) - (3 + 2i)|^2 = 8$, so $|(x - 3) + i(y - 2)|^2 = 8$. Using the definition of the modulus squared, $(x - 3)^2 + (y - 2)^2 = 8$.

Step 2: Analyzing Set $S_2$

The set $S_2$ is defined as $S_2 = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\}$.

• Explanation: We need to understand what this inequality represents in the complex plane. The real part of $z$ is $x$, so the inequality becomes $x \ge 5$.
• Applying the formula: Since $z = x + iy$, we have Re$(z) = x$. Thus, $x \ge 5$.
• Geometrical Interpretation: This represents the region in the complex plane to the right of and including the vertical line $x = 5$.

Step 3: Analyzing Set $S_3$

The set $S_3$ is defined as $S_3 = \{ z \in C||z - \overline z | \ge 8\}$.

• Explanation: We need to simplify the expression $|z - \overline{z}|$.
• Applying the formula: Since $z = x + iy$, the conjugate is $\overline{z} = x - iy$. Therefore, $z - \overline{z} = (x + iy) - (x - iy) = 2iy$. The inequality becomes $|2iy| \ge 8$.
• Simplifying the inequality: We have $|2iy| = 2|y|$, so $2|y| \ge 8$, which simplifies to $|y| \ge 4$.
• Geometrical Interpretation: The inequality $|y| \ge 4$ means $y \ge 4$ or $y \le -4$. This represents two horizontal regions: one above the line $y = 4$ (inclusive) and one below the line $y = -4$ (inclusive).

Step 4: Finding the Intersection $S_1 \cap S_2 \cap S_3$

We are looking for the complex numbers $z = x + iy$ that simultaneously satisfy the following conditions:

1. $(x - 3)^2 + (y - 2)^2 = 8$ (Circle)
2. $x \ge 5$ (Right half-plane)
3. $y \ge 4$ or $y \le -4$ (Two horizontal half-planes)

• Intersection of $S_1$ and $S_2$: We seek the portion of the circle $(x - 3)^2 + (y - 2)^2 = 8$ that lies in the region $x \ge 5$. To find the intersection of the circle and the line $x = 5$, we substitute $x = 5$ into the circle equation: $(5 - 3)^2 + (y - 2)^2 = 8$, which gives $4 + (y - 2)^2 = 8$, so $(y - 2)^2 = 4$. Taking the square root, $y - 2 = \pm 2$, so $y = 4$ or $y = 0$. Thus, the intersection points of the circle and the line $x=5$ are $(5, 4)$ and $(5, 0)$. For points on the circle with $x \ge 5$, the $y$-coordinate must lie between 0 and 4.

• Intersection with $S_3$: Now we intersect the result from above with $S_3$, which means we need to consider $y \ge 4$ or $y \le -4$. Since we know that $0 \le y \le 4$ from the intersection of $S_1$ and $S_2$, the only possibility is $y = 4$.

  Substitute $y = 4$ into the circle equation $(x - 3)^2 + (y - 2)^2 = 8$: $(x - 3)^2 + (4 - 2)^2 = 8$ $(x - 3)^2 + 4 = 8$ $(x - 3)^2 = 4$ $x - 3 = \pm 2$ $x = 5$ or $x = 1$.

  We have two candidate points: $(5, 4)$ and $(1, 4)$. We need to check both with all three conditions.

  • Candidate 1: $(5, 4)$

    • $S_1$: $(5 - 3)^2 + (4 - 2)^2 = 4 + 4 = 8$. (Satisfied)
    • $S_2$: $5 \ge 5$. (Satisfied)
    • $S_3$: $4 \ge 4$. (Satisfied) So, $z = 5 + 4i$ is a solution.

  • Candidate 2: $(1, 4)$

    • $S_1$: $(1 - 3)^2 + (4 - 2)^2 = 4 + 4 = 8$. (Satisfied)
    • $S_2$: $1 \ge 5$. (Not satisfied)
    • $S_3$: $4 \ge 4$. (Satisfied) Since condition $S_2$ is not satisfied, $z = 1 + 4i$ is not a solution.

Therefore, only one complex number, $z = 5 + 4i$, satisfies all three conditions.

Tips and Common Mistakes

• Visualizing: Drawing the circle, the vertical line $x = 5$, and the horizontal lines $y = 4$ and $y = -4$ helps in understanding the problem and identifying the regions of interest.
• Absolute Value: Remember that $|y| \ge 4$ means $y \ge 4$ OR $y \le -4$.
• Checking All Conditions: After finding candidate intersection points, always verify that they satisfy all the given conditions.

Summary

By analyzing each set and finding their intersection, we found that only one complex number, $z = 5 + 4i$, satisfies all three conditions. Therefore, the number of elements in $S_1 \cap S_2 \cap S_3$ is 1.

The final answer is \boxed{1}, which corresponds to option (A).