document.write( "Question 1066912: Find the greatest and least values of:
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Algebra.Com's Answer #682140 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "Find the greatest and least values of:
\n" ); document.write( "A) |z1+z2| where |z1|=6 and z2= 3+4i
\n" ); document.write( "B) Re(z) where |z-(2+i)| is less than or equal to 1.
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\n" ); document.write( "\n" ); document.write( "I will answer the first question A) only.\r
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document.write( "1.  Imagine the point z2 = 3 + 4i in the complex plane.\r\n" );
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document.write( "    By the way, its modulus is  = 5.\r\n" );
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document.write( "2.  The set of points {z | |z| = 6} is the circle on the complex plane of the radius 6 with the center at the origin.\r\n" );
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document.write( "3.  The set of complex numbers {z | z = z1 + z2} is the circle on the complex plane of the radius 6 with the center at the point z2.\r\n" );
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document.write( "4.  The modulus |z1 + z2| is maximal when the vector z1 is collinear (parallel) to the vector z2. \r\n" );
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document.write( "    It means z1 = .\r\n" );
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document.write( "    If so, then the maximum of |z1 + z2| is equal to 5 + 6 = 11.\r\n" );
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document.write( "5.  The modulus |z1 + z2| is minimal when the vector z1 is opposite (anti-parallel) to the vector z2. \r\n" );
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document.write( "    It means z1 = .\r\n" );
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document.write( "    If so, then the minimum of |z1 + z2| is equal to |5 - 6| = |-1| = 1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Question A) is answered.\r
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\n" ); document.write( "\n" ); document.write( "To understand my writing, you must freely play with all related notions.
\n" ); document.write( "It is not, actually, the requirement from my side.
\n" ); document.write( "It is what the problem does require.\r
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\n" ); document.write( "\n" ); document.write( "On complex numbers, see the lessons \r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetic operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Complex numbers\".\r
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