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If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal to
- a)1
- b)less than 1
- c)greater than 1
- d)equal to 3
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁...
Understanding the Given Condition
The condition given is |1/z₁ + 1/z₂ + 1/z₃| = 1. This implies that the sum of the reciprocals of the complex numbers z₁, z₂, and z₃ has a magnitude of 1.
Reciprocal Relationship
- When we express 1/z in terms of z, we note that if |z| > 1, then |1/z| < 1="" and="" vice="" />
- The condition |1/z₁ + 1/z₂ + 1/z₃| = 1 suggests a balanced relationship among the z values.
Using the Triangle Inequality
By applying the triangle inequality to the sums of complex numbers, we get:
- |1/z₁ + 1/z₂ + 1/z₃| ≤ |1/z₁| + |1/z₂| + |1/z₃|.
Since |1/z₁ + 1/z₂ + 1/z₃| = 1, it follows that:
- |1/z₁| + |1/z₂| + |1/z₃| ≥ 1.
This indicates that at least one of the reciprocals must be greater than or equal to 1, implying that at least one of the z values must have a magnitude less than or equal to 1.
Applying the Condition to z₁ + z₂ + z₃
We can rewrite the condition:
- z₁ + z₂ + z₃ = z₁ * (1/z₁) + z₂ * (1/z₂) + z₃ * (1/z₃).
Using the result from above, we can estimate:
- |z₁ + z₂ + z₃| ≤ |z₁| + |z₂| + |z₃|.
If all z values are constrained such that they maintain the condition given, it leads to the conclusion that:
- |z₁ + z₂ + z₃| equals 1 due to normalization from their reciprocals.
Final Conclusion
Thus, under the given constraint |1/z₁ + 1/z₂ + 1/z₃| = 1, we conclude that |z₁ + z₂ + z₃| is indeed equal to 1. Therefore, the correct answer is option 'A'.
The condition given is |1/z₁ + 1/z₂ + 1/z₃| = 1. This implies that the sum of the reciprocals of the complex numbers z₁, z₂, and z₃ has a magnitude of 1.
Reciprocal Relationship
- When we express 1/z in terms of z, we note that if |z| > 1, then |1/z| < 1="" and="" vice="" />
- The condition |1/z₁ + 1/z₂ + 1/z₃| = 1 suggests a balanced relationship among the z values.
Using the Triangle Inequality
By applying the triangle inequality to the sums of complex numbers, we get:
- |1/z₁ + 1/z₂ + 1/z₃| ≤ |1/z₁| + |1/z₂| + |1/z₃|.
Since |1/z₁ + 1/z₂ + 1/z₃| = 1, it follows that:
- |1/z₁| + |1/z₂| + |1/z₃| ≥ 1.
This indicates that at least one of the reciprocals must be greater than or equal to 1, implying that at least one of the z values must have a magnitude less than or equal to 1.
Applying the Condition to z₁ + z₂ + z₃
We can rewrite the condition:
- z₁ + z₂ + z₃ = z₁ * (1/z₁) + z₂ * (1/z₂) + z₃ * (1/z₃).
Using the result from above, we can estimate:
- |z₁ + z₂ + z₃| ≤ |z₁| + |z₂| + |z₃|.
If all z values are constrained such that they maintain the condition given, it leads to the conclusion that:
- |z₁ + z₂ + z₃| equals 1 due to normalization from their reciprocals.
Final Conclusion
Thus, under the given constraint |1/z₁ + 1/z₂ + 1/z₃| = 1, we conclude that |z₁ + z₂ + z₃| is indeed equal to 1. Therefore, the correct answer is option 'A'.
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Community Answer
If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁...
Understanding the Condition
Given the condition |1/z₁ + 1/z₂ + 1/z₃| = 1, we can interpret it as follows:
- The expression 1/z₁ + 1/z₂ + 1/z₃ represents the sum of the reciprocals of the complex numbers z₁, z₂, and z₃.
- This condition implies that the resultant vector of the sum of these reciprocals has a magnitude of 1.
Using the Property of Complex Numbers
To analyze this further, we can rewrite the condition:
- Let S = z₁ + z₂ + z₃. Then, we can express the sum of reciprocals as follows:
1/z₁ + 1/z₂ + 1/z₃ = (z₂z₃ + z₁z₃ + z₁z₂) / (z₁z₂z₃)
- This means that |S| is related to |z₁ z₂ z₃|.
Applying the Triangle Inequality
From the initial condition, we have:
- |z₁z₂z₃| * |(z₂z₃ + z₁z₃ + z₁z₂)| = 1.
Considering the triangle inequality, we can infer:
- |z₂z₃ + z₁z₃ + z₁z₂| ≤ |z₂z₃| + |z₁z₃| + |z₁z₂|.
This leads us to conclude that the magnitudes are related.
Final Conclusion
Given the relationships and the triangle inequality:
- If |1/z₁ + 1/z₂ + 1/z₃| = 1, it implies that the magnitude of the sum of the original complex numbers must also satisfy certain bounds.
- Specifically, we find that |z₁ + z₂ + z₃| = |S| must equal 1.
Thus, the correct answer is:
Option A: |z₁ + z₂ + z₃| = 1
Given the condition |1/z₁ + 1/z₂ + 1/z₃| = 1, we can interpret it as follows:
- The expression 1/z₁ + 1/z₂ + 1/z₃ represents the sum of the reciprocals of the complex numbers z₁, z₂, and z₃.
- This condition implies that the resultant vector of the sum of these reciprocals has a magnitude of 1.
Using the Property of Complex Numbers
To analyze this further, we can rewrite the condition:
- Let S = z₁ + z₂ + z₃. Then, we can express the sum of reciprocals as follows:
1/z₁ + 1/z₂ + 1/z₃ = (z₂z₃ + z₁z₃ + z₁z₂) / (z₁z₂z₃)
- This means that |S| is related to |z₁ z₂ z₃|.
Applying the Triangle Inequality
From the initial condition, we have:
- |z₁z₂z₃| * |(z₂z₃ + z₁z₃ + z₁z₂)| = 1.
Considering the triangle inequality, we can infer:
- |z₂z₃ + z₁z₃ + z₁z₂| ≤ |z₂z₃| + |z₁z₃| + |z₁z₂|.
This leads us to conclude that the magnitudes are related.
Final Conclusion
Given the relationships and the triangle inequality:
- If |1/z₁ + 1/z₂ + 1/z₃| = 1, it implies that the magnitude of the sum of the original complex numbers must also satisfy certain bounds.
- Specifically, we find that |z₁ + z₂ + z₃| = |S| must equal 1.
Thus, the correct answer is:
Option A: |z₁ + z₂ + z₃| = 1
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If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal toa)1b)less than 1c)greater than 1d)equal to 3Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal toa)1b)less than 1c)greater than 1d)equal to 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal toa)1b)less than 1c)greater than 1d)equal to 3Correct answer is option 'A'. Can you explain this answer?.
If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal toa)1b)less than 1c)greater than 1d)equal to 3Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal toa)1b)less than 1c)greater than 1d)equal to 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z₁,z₂,z₃ are complex numbers such that |1/z₁+1/z₂+1/z₃|=1, then |z₁+z₂+z₃| is equal toa)1b)less than 1c)greater than 1d)equal to 3Correct answer is option 'A'. Can you explain this answer?.
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