The number of elements in the set $\{z=a+ib \in \mathbb{C}: a, b \in \mathbb{Z}$ and $1<|z-3+2i|<4\}$ is ____.

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Complex Numbers

The number of elements in the set

{z = a + ib \in C : a, b \in Z

and

1 < ∣ z - 3 + 2 i ∣ < 4}

is ____.

Answer: 40

We need complex numbers

z = a + ib

with integer coordinates satisfying:

1 < ∣ z - 3 + 2 i ∣ < 4

1 < ∣ (a - 3) + i (b + 2) ∣ < 4

1 < (a - 3)^{2} + (b + 2)^{2} < 16

This represents lattice points strictly between two circles centered at

(3, - 2)

with radii 1 and 4.

We need

(a - 3)^{2} + (b + 2)^{2} \in {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

.

Systematically counting integer points:

Total: 40 lattice points satisfy the condition.

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Integer answer