Let z1,z2,z3,z4 be complex numbers satisfying 1≤|z1−3−4i|≤2, 3≤|z2+6−8i|≤4,|z3|=|z4|=1. Choose the correct statements

∴ 1⩽||z1|−5|⩽|z1−3−4i|≤2 (|z1|−5)∈ [−2,−1]∪[1,2] |z1|∈[3,4]∪[6,7] similarly, 3⩽||z2|−10|≤|z2+6−8i|≤4 (|z2|−10)∈[−4,−3]∪[3,4] |z2|∈[6,7]∪[13,14] |z3−2z1|≤|z3|+2|z1||z3−2z1|≤1+2.7=15 |z2−z4|≥||z2|−|z4||=||z2|−1|=5

Let z1,z2,z3,z4 be complex numbers satisfying 1≤|z1−3−4i|≤2, 3≤|z2+6−8i|≤4,|z3|=|z4|=1. Choose the correct statements

∴ 1⩽||z1|−5|⩽|z1−3−4i|≤2 (|z1|−5)∈ [−2,−1]∪[1,2] |z1|∈[3,4]∪[6,7] similarly, 3⩽||z2|−10|≤|z2+6−8i|≤4 (|z2|−10)∈[−4,−3]∪[3,4] |z2|∈[6,7]∪[13,14] |z3−2z1|≤|z3|+2|z1||z3−2z1|≤1+2.7=15 |z2−z4|≥||z2|−|z4||=||z2|−1|=5

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Let $z_{1}, z_{2}, z_{3}, z_{4}$ be complex numbers satisfying
$1 \leq | z_{1} - 3 - 4 i | \leq 2, 3 \leq | z_{2} + 6 - 8 i | \leq 4,$ $| z_{3} | = | z_{4} | = 1$ .
Choose the correct statements

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Range of $| z_{1} |$ is $[3, 5] \cup [6, 8]$

Range of $| z_{2} |$ is $[6, 7] \cup [13, 14]$

Maximum possible value of $| z_{3} - 2 z_{1} |$ is less than or equal to 15

Minimum possible value of $| z_{2} - z_{4} |$ is greater than or equal to 5

answer is C, D.

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Detailed Solution

$∴ 1 ⩽ | | z_{1} | - 5 | ⩽ | z_{1} - 3 - 4 i | \leq 2 (| z_{1} | - 5) \in [- 2, - 1] \cup [1, 2] | z_{1} | \in [3, 4] \cup [6, 7]$
similarly, $3 ⩽ | | z_{2} | - 10 | \leq | z_{2} + 6 - 8 i | \leq 4$
$(| z_{2} | - 10) \in [- 4, - 3] \cup [3, 4] | z_{2} | \in [6, 7] \cup [13, 14] \begin{array}{l} | z_{3} - 2 z_{1} | \leq | z_{3} | + 2 | z_{1} | \\ | z_{3} - 2 z_{1} | \leq 1 + 2.7 = 15 \end{array} | z_{2} - z_{4} | \geq | | z_{2} | - | z_{4} | | = | | z_{2} | - 1 | = 5$