Question

Let $A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \}$, $B = \{ z \in \mathbb{C} : \text{Re}(z - iz) = 2 \}$, and $S = A \cap B$. Then $\sum_{z \in S} |z|^2$ is equal to

Hint:

Solve the system of equations to find the intersection points of the sets.

Answer 1

Correct Answer: 22

1. Identify the sets $A$ and $B$: - $A: |z - 2 - i| = 3$ \[ |(x - 2) + (y - 1)i| = 3 \] \[ (x - 2)^2 + (y - 1)^2 = 9 \] - $B: \text{Re}(z - iz) = 2$ \[ \text{Re}((x + y) + i(y - x)) = 2 \] \[ x + y = 2 \]
2. Solve the system of equations: \[ \begin{cases} (x - 2)^2 + (y - 1)^2 = 9 x + y = 2 \end{cases} \] - Substitute $y = 2 - x$ into the first equation: \[ (x - 2)^2 + (2 - x - 1)^2 = 9 \] \[ (x - 2)^2 + (1 - x)^2 = 9 \] \[ x^2 - 4x + 4 + 1 - 2x + x^2 = 9 \] \[ 2x^2 - 6x + 5 = 9 \] \[ 2x^2 - 6x - 4 = 0 \] \[ x^2 - 3x - 2 = 0 \] \[ x = \frac{3 \pm \sqrt{17}}{2} \] - Corresponding $y$ values: \[ y = 2 - x = 2 - \frac{3 \pm \sqrt{17}}{2} = \frac{1 \mp \sqrt{17}}{2} \]
3. Calculate $\sum_{z \in S} |z|^2$: \[ \sum_{z \in S} |z|^2 = \left( \frac{3 + \sqrt{17}}{2} \right)^2 + \left( \frac{1 - \sqrt{17}}{2} \right)^2 + \left( \frac{3 - \sqrt{17}}{2} \right)^2 + \left( \frac{1 + \sqrt{17}}{2} \right)^2 \] \[ = \frac{1}{4} \left[ 2 \times 26 + 2 \times 18 \right] = \frac{88}{4} = 22 \] Therefore, the correct answer is (1) 22.

Answer 2

We have:

A: |z − (2+i)| = 3 ⇒ (x−2)^2 + (y−1)^2 = 9 (circle)

B: Re(z − iz) = 2 ⇒ Re((1−i)(x+iy)) = x + y = 2 (line)

Step-by-Step Solution:

Step 1: Intersect the circle with the line x + y = 2. Put y = 2 − x into the circle:

\[ (x-2)^2 + (1 - x)^2 = 9 \;\Longrightarrow\; 2x^2 - 6x + 5 = 9 \] \[ \Longrightarrow\; 2x^2 - 6x - 4 = 0 \;\Longrightarrow\; x^2 - 3x - 2 = 0. \]

Thus roots \(x_1, x_2\) satisfy \(x_1 + x_2 = 3,\; x_1 x_2 = -2\), and \(y_i = 2 - x_i\).

Step 2: For each point \(z_i = x_i + i y_i\),

\[ |z_i|^2 = x_i^2 + y_i^2 = (x_i + y_i)^2 - 2x_i y_i = 4 - 2x_i y_i \] (since \(x_i + y_i = 2\)).

 

Step 3: Sum over the two intersection points:

\[ \sum |z_i|^2 = \sum (4 - 2x_i y_i) = 8 - 2\sum x_i y_i. \] \[ x_i y_i = x_i (2 - x_i) = 2x_i - x_i^2 \Rightarrow \sum x_i y_i = 2(x_1+x_2) - (x_1^2 + x_2^2). \] \[ x_1^2 + x_2^2 = (x_1+x_2)^2 - 2x_1x_2 = 3^2 - 2(-2) = 9 + 4 = 13. \] \[ \sum x_i y_i = 2\cdot 3 - 13 = -7. \] \[ \therefore\ \sum |z|^2 = 8 - 2(-7) = 8 + 14 = \boxed{22}. \]