SOLUTION: Find the greatest and least values of: A) |z1+z2| where |z1|=6 and z2= 3+4i B) Re(z) where |z-(2+i)| is less than or equal to 1.

Question 1066912: Find the greatest and least values of:
A) |z1+z2| where |z1|=6 and z2= 3+4i
B) Re(z) where |z-(2+i)| is less than or equal to 1.
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
Find the greatest and least values of:
A) |z1+z2| where |z1|=6 and z2= 3+4i
B) Re(z) where |z-(2+i)| is less than or equal to 1.
~~~~~~~~~~~~~~~~~~~~~

I will answer the first question A) only.

1. Imagine the point z2 = 3 + 4i in the complex plane. By the way, its modulus is = 5. 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. 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. 4. The modulus |z1 + z2| is maximal when the vector z1 is collinear (parallel) to the vector z2. It means z1 = . If so, then the maximum of |z1 + z2| is equal to 5 + 6 = 11. 5. The modulus |z1 + z2| is minimal when the vector z1 is opposite (anti-parallel) to the vector z2. It means z1 = . If so, then the minimum of |z1 + z2| is equal to |5 - 6| = |-1| = 1.

Question A) is answered.

To understand my writing, you must freely play with all related notions.
It is not, actually, the requirement from my side.
It is what the problem does require.

On complex numbers, see the lessons
    - Complex numbers and arithmetic operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Complex numbers".

Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
A) The greatest and least values of
will happen when the angles for the polar form of and
are the same or apart.
When that happens,
or

In this case:
,
so the greatest and least values asked for are
and

B) Let it be
, where and are real numbers.

means
Squaring both sides of the equal sign in that real numbers equation
gives us the real numbers equation
,
which contains all the solutions for ,
and could have new, extraneous solutions only if .
Since for real numbers,
there will be no extraneous solutions.
All we have to do is solve for .
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