Let A, B, C three set of complex number defined belowA={z:|z+1|≤2+Re⁡(z)};B={z:|z−1|≥1};Cz:z−1z+1≥1The number of points having integral coordinate in region A∩B∩C is

|z+1|≤2+Re⁡(z)x+12+y2≤2+xx2+2x+1+y2≤4+x2+4x⇒y2≤3x+3|z−1|≥1x-12+y2≥1⇒x2+y2-2x≥0z−1z+1≥1⇒z−1≥z+1x-12+y2≥x+12+y2⇒-4x≥0⇒x≤0A:y2≤3(x+1);B:(x+1)2+y2≥1;C:x≥0

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Let A, B, C three set of complex number defined below
$A = {z : | z + 1 | \leq 2 + Re (z)}; B = {z : | z - 1 | \geq 1}; C \{z : |\frac{z - 1}{z + 1}| \geq 1\}$
The number of points having integral coordinate in region $A \cap B \cap C$ is

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Detailed Solution

$| z + 1 | \leq 2 + Re (z)$

$\sqrt{{(x + 1)}^{2} + y^{2}} \leq 2 + x$

$(x^{2} + 2 x + 1) + y^{2} \leq 4 + x^{2} + 4 x \Rightarrow y^{2} \leq 3 x + 3$

$| z - 1 | \geq 1$

${(x - 1)}^{2} + y^{2} \geq 1 \Rightarrow x^{2} + y^{2} - 2 x \geq 0$

$|\frac{z - 1}{z + 1}| \geq 1 \Rightarrow |z - 1| \geq |z + 1|$

${(x - 1)}^{2} + y^{2} \geq {(x + 1)}^{2} + y^{2} \Rightarrow - 4 x \geq 0 \Rightarrow x \leq 0$

$A : y^{2} \leq 3 (x + 1); B : (x + 1)^{2} + y^{2} \geq 1; C : x \geq 0$

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