The number of elements in the set z = a + i b ∈ ℂ : a , b ∈ ℤ and 1 < z - 3 + 2 i < 4 is _____.

Step-by-step solution

We know that z - 3 - 2 i = r represents a circle with centre 3 , - 2 and radius r . 1 < z - 3 + 2 i < 4 1 < a - 3 2 + b + 2 2 < 16 The ordered pairs of a , b satisfying the above inequality are 0 , ± 2 , ± 2 , 0 , ± 1 , ± 2 , ± 2 , ± 1 ± 2 , ± 3 , 3 ± , ± 2 , ± 1 , ± 1 , 2 ± , ± 2 ± 3 , 0 , 0 , ± 3 , ± 3 ± 1 , ± 1 , ± 3 i.e. total 40 elements are present in the given set