complex number euler form modulus question (1 Viewer)

gamja

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How would you solve this? Apparently there are methods using geometrical graphing or using trig algebra - feel free to solve outside these methods tho.

Much thanks! :D
 

yanujw

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The most prominent method is to let . Then use the fact that

The locus of z itself is an ellipse which would have been helpful in the old syllabus. However, there is still a geometric way to find the maximum value.

Recall that means the combined distance of from the points 2 and -2 on the Argand diagram. And represents the unit circle of points. The combined length of the two vectors as visualised below is maximised* at either the points i or -i (see diagram below). At these points, each line has a length of so the maximum combined length is

*this is just a vague conclusion based on looking at the graph, it may require an actual proof which just goes back to the algebraic version anyways.

1678062414604.png
 
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Drongoski

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The most prominent method is to let . Then use the fact that

The locus of z itself is an ellipse which would have been helpful in the old syllabus. However, there is still a geometric way to find the maximum value.

Recall that means the combined distance of from the points 2 and -2 on the Argand diagram. And represents the unit circle of points. The combined length of the two vectors as visualised below is maximised* at either the points i or -i (see diagram below). At these points, each line has a length of so the maximum combined length is

*this is just a vague conclusion based on looking at the graph, it may require an actual proof which just goes back to the algebraic version anyways.

View attachment 37914
The geometric method (see your diagram) is the easiest to understand and the shortest.
 

tywebb

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Some dude (can't remember who) came here before and tried to use the AM-GM inequality to do it but it didn't work. They kind of had a good idea but had the inequality the wrong way around.

However the AM-QM inequality does work (instead of the AM-GM one):

 
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member 6003

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Some dude (can't remember who) came here before and tried to use the AM-GM inequality to do it but it didn't work. They kind of had a good idea but had the inequality the wrong way around.

However the AM-QM inequality does work (instead of the AM-GM one):

I was the dude, cool solution but ofc its outside the syllabus. I saw 2sqrt(5) in my solution so I just thought it would be correct without considering any logic before posting.
 

tywebb

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I don't think it is outside the syllabus if you prove this first (pretty easy to do):

 

member 6003

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well if you prove first then yea they couldn't mark you down for it
 

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