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If Z is a complex number, then the minimum value of |z| + |z-l| is
- a)1
- b)0
- c)1/2
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b...
Note that
, hence minimum value is 1 and it is attained at Z = 0, 1/2
, hence minimum value is 1 and it is attained at Z = 0, 1/2
Most Upvoted Answer
If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b...
Note that
, hence minimum value is 1 and it is attained at Z = 0, 1/2
, hence minimum value is 1 and it is attained at Z = 0, 1/2
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Community Answer
If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b...
Explanation:
Definition:
- The modulus of a complex number z = x + yi is defined as |z| = sqrt(x^2 + y^2).
- |z-l| represents the modulus of the difference between z and l, where l is another complex number.
Minimum value of |z| + |z-l|:
- Let z = x + yi, where x and y are real numbers.
- Substituting z into the expression |z| + |z-l| gives us |x + yi| + |x-l + yi|.
- Using the definition of modulus, this expression becomes sqrt(x^2 + y^2) + sqrt((x-l)^2 + y^2).
Minimum value analysis:
- To find the minimum value of the expression sqrt(x^2 + y^2) + sqrt((x-l)^2 + y^2), we can use the triangle inequality.
- The triangle inequality states that for any two complex numbers z1 and z2, |z1 + z2| <= |z1|="" +="">
- Applying the triangle inequality to our expression, we get |z| + |z-l| >= |z - (z-l)| = |l|.
- Therefore, the minimum value of |z| + |z-l| is |l|.
Conclusion:
- In this case, the minimum value of |z| + |z-l| is |l|.
- Hence, the correct answer is option 'A' (1).
Definition:
- The modulus of a complex number z = x + yi is defined as |z| = sqrt(x^2 + y^2).
- |z-l| represents the modulus of the difference between z and l, where l is another complex number.
Minimum value of |z| + |z-l|:
- Let z = x + yi, where x and y are real numbers.
- Substituting z into the expression |z| + |z-l| gives us |x + yi| + |x-l + yi|.
- Using the definition of modulus, this expression becomes sqrt(x^2 + y^2) + sqrt((x-l)^2 + y^2).
Minimum value analysis:
- To find the minimum value of the expression sqrt(x^2 + y^2) + sqrt((x-l)^2 + y^2), we can use the triangle inequality.
- The triangle inequality states that for any two complex numbers z1 and z2, |z1 + z2| <= |z1|="" +="">
- Applying the triangle inequality to our expression, we get |z| + |z-l| >= |z - (z-l)| = |l|.
- Therefore, the minimum value of |z| + |z-l| is |l|.
Conclusion:
- In this case, the minimum value of |z| + |z-l| is |l|.
- Hence, the correct answer is option 'A' (1).
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If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b)0c)1/2d)None of theseCorrect answer is option 'A'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b)0c)1/2d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b)0c)1/2d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b)0c)1/2d)None of theseCorrect answer is option 'A'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b)0c)1/2d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If Z is a complex number, then the minimum value of |z| + |z-l| isa)1b)0c)1/2d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
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