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Question
For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, the minimum value of |z1 - z2| is ______.
Options
0
2
7
17
MCQ
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Solution
For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, the minimum value of |z1 - z2| is 2.
Explanation:
We have,
|z1| = |z2 - (3 + 4i) + 3 + 4i|
⇒ |z1| ≤ |z2 - (3 + 4i)| + |3 + 4i|
⇒ |z1| ≤ 5 + 5 ....[∵ |3 + 4i| = `sqrt(9 + 16)` = 5]
⇒ |z1| ≤ 10 ⇒ - |z2| ≥ - 10
⇒ |z1| - |z2| ≥ |z1| - 10
⇒ |z1| - |z2| ≥ 12 - 10
⇒ |z1| - |z2| ≥ 2
⇒ |z1 - z2| ≥ 2 .....[∵ |z1 - z2| ≥ |z1| - |z2|]
∴ minimum value of |z1 - z2| = 2
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Argand Diagram Or Complex Plane
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