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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?
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If z is any complex number satisfying |z – 3 – 2i| < 2,...
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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.
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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?
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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?.
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