Question

The sum of the square of the modulus of the elements in the set $\\{z = a + ib : a, b \in \mathbb{Z}, z \in \mathbb{C}, |z-1| \leq 1, |z-5| \leq |z-5i|\\}$ is ________.

Answer 1

Step 1: Analyze the conditions

The first condition is:

$|z - 1| \leq 1$.

Substitute $z = x + iy$, where $x, y \in \mathbb{R}$:

$|z - 1| = \sqrt{(x - 1)^2 + y^2} \leq 1$.

Squaring both sides:

$(x - 1)^2 + y^2 \leq 1.$ (1)

The second condition is:

$|z - 5| \leq |z - 5i|.$

Substitute $z = x + iy$:

$\sqrt{(x - 5)^2 + y^2} \leq \sqrt{x^2 + (y - 5)^2}.$

Squaring both sides:

$(x - 5)^2 + y^2 \leq x^2 + (y - 5)^2.$

Simplify:

$-10x - 10y \leq 0.$

$x + y \geq 0.$ (2)

Step 2: Solve the inequalities

From condition (1), $(x - 1)^2 + y^2 \leq 1$, the points lie within or on a circle centered at $(1, 0)$ with radius 1.

From condition (2), $x + y \geq 0$, the points lie above or on the line $y = -x$.

Step 3: Discretize $x, y$ as integers

Since $x, y \in \mathbb{Z}$, we identify the integer points satisfying both conditions. These points are:

$(0, 0), (1, 0), (2, 0), (1, 1), (1, -1).$

Step 4: Compute the square of the modulus

For each point $z_k = x_k + iy_k$, the modulus squared is $|z_k|^2 = x_k^2 + y_k^2$. Calculate for each point:

• $|z_1|^2 = |0 + 0i|^2 = 0^2 + 0^2 = 0,$
• $|z_2|^2 = |1 + 0i|^2 = 1^2 + 0^2 = 1,$
• $|z_3|^2 = |2 + 0i|^2 = 2^2 + 0^2 = 4,$
• $|z_4|^2 = |1 + i|^2 = 1^2 + 1^2 = 2,$
• $|z_5|^2 = |1 - i|^2 = 1^2 + (-1)^2 = 2.$

Step 5: Sum the squares of the modulus

$\sum_{k=1}^{5} |z_k|^2 = 0 + 1 + 4 + 2 + 2 = 9.$

Final Answer is $9$.