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

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

JEE MainJEE Main 2022Complex Numbers and Quadratic Equations

Solution:

$1 < |Z-3+2 i| < 4$
image
$1 < (a-3)^{2}+(b+2)^{2} < 16$
$(0, \pm 2),(\pm 2,0),(\pm 1, \pm 2),(\pm 2, \pm 1) $
$(\pm 2, \pm 3),(3 \pm, \pm 2),(\pm 1, \pm 1),(2 \pm, \pm 2) $
$(\pm 3,0),(0, \pm 3),(\pm 3 \pm 1),(\pm 1, \pm 3)$
Total $40$ points