JEE Exam > JEE Questions > If z is any complex number satisfying |z &nda...
Start Learning for Free
If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| is
Correct answer is '5'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
If z is any complex number satisfying |z – 3 – 2i| < 2,...
Most Upvoted Answer
If z is any complex number satisfying |z – 3 – 2i| < 2,...
Since |z| < 1,="" we="" have="" |z|^2="" />< 1.="" />
Let w = 1 - z. Then
|w| = |1 - z| = |z - 1|
and
|w|^2 = |1 - z|^2 = |z - 1|^2 = (z - 1)(\overline{z} - 1)
= |z|^2 - z - \overline{z} + 1
< 1="" -="" z="" -="" \overline{z}="" +="" 1="2" -="" (z="" +="" \overline{z})="" />
= 2 - 2\operatorname{Re}(z)
≤ 2.
(Note that we used the fact that z + \overline{z} = 2\operatorname{Re}(z) for any complex number z.)
Since |w|^2 < 2,="" it="" follows="" that="" |w|="" />< \sqrt{2}.="" />
Therefore, we have
|1 - z| < \sqrt{2},="" />
which implies that the point representing z lies inside the open disk of radius \sqrt{2} centered at 1.
Let w = 1 - z. Then
|w| = |1 - z| = |z - 1|
and
|w|^2 = |1 - z|^2 = |z - 1|^2 = (z - 1)(\overline{z} - 1)
= |z|^2 - z - \overline{z} + 1
< 1="" -="" z="" -="" \overline{z}="" +="" 1="2" -="" (z="" +="" \overline{z})="" />
= 2 - 2\operatorname{Re}(z)
≤ 2.
(Note that we used the fact that z + \overline{z} = 2\operatorname{Re}(z) for any complex number z.)
Since |w|^2 < 2,="" it="" follows="" that="" |w|="" />< \sqrt{2}.="" />
Therefore, we have
|1 - z| < \sqrt{2},="" />
which implies that the point representing z lies inside the open disk of radius \sqrt{2} centered at 1.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer?
Question Description
If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer?.
If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer?.
Solutions for If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer?, a detailed solution for If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? has been provided alongside types of If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If z is any complex number satisfying |z – 3 – 2i| < 2, then the minimum value of |2z – 6 + 5i| isCorrect answer is '5'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup