Question

Let C be the set of all complex numbers. Let $S_1 = \{z \in C : |z-3-2i|^2 = 8\}$, $S_2 = \{z \in C : \text{Re}(z) \ge 5\}$ and $S_3 = \{z \in C : |z - \bar{z}| \ge 8\}$. Then the number of elements in $S_1 \cap S_2 \cap S_3$ is equal to :

• A. 0
• B. 1
• C. 2
• D. Infinite

Hint:

For intersection problems in the complex plane, always convert conditions into Cartesian form and compare ranges. Boundary analysis often reveals whether solutions are finite, infinite, or empty.

Answer 1

The Correct Option is B

Let $z = x + iy$, where $x,y \in \mathbb{R}$. Step 1: Interpretation of each set Set $S_1$: \[ |z - (3+2i)|^2 = 8 \] \[ |(x-3) + i(y-2)|^2 = 8 \Rightarrow (x-3)^2 + (y-2)^2 = 8 \] This represents a circle with: \[ \text{Center } (3,2), \quad \text{Radius } \sqrt{8} = 2\sqrt{2} \] Set $S_2$: \[ \text{Re}(z) \ge 5 \Rightarrow x \ge 5 \] This is the closed half-plane to the right of the vertical line $x=5$. Set $S_3$: \[ |z - \bar{z}| \ge 8 \] \[ |(x+iy)-(x-iy)| = |2iy| = 2|y| \ge 8 \Rightarrow |y| \ge 4 \] This corresponds to the region: \[ y \ge 4 \quad \text{or} \quad y \le -4 \] Step 2: Geometric feasibility For the circle $(x-3)^2 + (y-2)^2 = 8$: \[ x \in [3-2\sqrt{2},\, 3+2\sqrt{2}] \approx [0.17,\, 5.83] \] \[ y \in [2-2\sqrt{2},\, 2+2\sqrt{2}] \approx [-0.83,\, 4.83] \] Since the circle does not extend below $y=-4$, only the region $y \ge 4$ from $S_3$ is relevant. Step 3: Boundary checking If $x>5$ and $y>4$, then: \[ (x-3)^2>4,\quad (y-2)^2>4 \Rightarrow (x-3)^2+(y-2)^2>8 \] So no such point lies on the circle. Hence, possible solutions must lie on the boundaries: \[ x=5 \quad \text{or} \quad y=4 \] Case 1: $x=5$ \[ (5-3)^2 + (y-2)^2 = 8 \Rightarrow 4 + (y-2)^2 = 8 \Rightarrow (y-2)^2 = 4 \] \[ y = 4 \text{ or } 0 \] Point $(5,4)$ satisfies: \[ x \ge 5,\quad |y| \ge 4 \] ✔ Valid Point $(5,0)$ fails $|y| \ge 4$ ✘ Invalid Case 2: $y=4$ \[ (x-3)^2 + (4-2)^2 = 8 \Rightarrow (x-3)^2 = 4 \Rightarrow x = 5 \text{ or } 1 \] Point $(5,4)$ already counted ✔ Valid Point $(1,4)$ fails $x \ge 5$ ✘ Invalid Step 4: Conclusion Only one complex number satisfies all three conditions: \[ z = 5 + 4i \] \[ \boxed{\text{Number of elements } = 1} \]