Math Is Fun Forum

sunflower84

Member

Registered: 2009-05-27

Posts: 2

finding the maximum of arg z

Given that u=1+2i, and also the |z-u|=2.

Already sketch the locus of the complex number z such that|z-u|=2.

Now, question ask me to find the greatest value of arg z for points on this locus.

How can I get the answer?

Thx for helping ya.!:)

Last edited by sunflower84 (2009-05-28 03:05:13)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: finding the maximum of arg z

I was about to attack this with horrible-looking calculus, but then I saw a much nicer way.

You should be able to find the minimum of arg(z) without much trouble.
Once you have that, draw a line of symmetry between 0 and u, and that will get you the maximum of arg(z) without much work.

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Why did the vector cross the road?
It wanted to be normal.

sunflower84

Member

Registered: 2009-05-27

Posts: 2

Re: finding the maximum of arg z

ok......by following your way,
i can get the correct answer as shown in the book.
but i think i need a little bit more expaination on this.
Why the maximum of arg(z) is the double of the arg(u)?
_______________________________________________

Looking for your explaination......
thank you so much!!

Last edited by sunflower84 (2009-05-28 03:20:32)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: finding the maximum of arg z

That happens because the minimum argument is 0.
By symmetry, it's required that max(arg(z)) - arg(u) = arg(u) - min(arg(z)).

(That's not a general law, but it's easy to see why it's true here if you consider it geometrically)

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Why did the vector cross the road?
It wanted to be normal.