Question

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

Answer 1

At line y = –2, we have (5, –2) (6, –2) (1, –2) (0, –2)
⇒ 4 points
At line y = –1, we have (4, –1) (5, –1) (6, –1) (2, –1) (1, –1) (0, –1)
⇒ 6 points
At line y = 0, we have (0, 0) (1, 0) (2, 0) (3, 0) (4, 0) (5, 0) (6, 0)
⇒ 7 points
At line y = 1, we have (1, 1), (2, 1), (3, 1), (4, 1), (5, 1) i.e. 5 points
Similarly,
At line y = –5, we have 5 points
At line y = –4, we have 7 points
At line y = –3, we have 6 points
Then, the total integral points = 2(5 + 7 + 6) + 4 = 40

So, the answer is 40.