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if |z|>= 3 the least value of |z + 1/z| is?
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hartnn
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i think we need to use
|x|+|y|>|x+y|
but not sure.
Yahoo!
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But wat Does that Least Value For?
hartnn
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i m just trying....
if |z|>=3 what can u say about |1/z| ??
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|z -(- 1/z)| > = |z| - |-1/z|
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I Did nt Understand....wat is here Least..""
hartnn
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means |z+1/z| is > = 'some number'
u need to find that number.
hartnn
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but |z+1/z| <= |z|+|1/z|
how did u get > sign there ?
Yahoo!
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u see..i have inserted a -ve there....Since it asks for least value."
Callisto
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Hmm.. Am I right for this:
|z| ≥ 3
i.e. z ≥ 3
or z ≤ -3
?
hartnn
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i think z is complex here, isn't it @Yahoo! ??
Yahoo!
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Yes!
hartnn
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so |z|>= 3 represents region outside the circle with radius 3.....if i have not mis-interpreted
Callisto
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Wow! Something I don't know! I'm sorry!!
Yahoo!
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"OUTSIDE" hw
hartnn
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because of > sign, if < then inside.
Yahoo!
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Ok...)
Yahoo!
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@hartnn Any Idea....
Yahoo!
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i think we have to find the least dist b/w Circles
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@hartnn i Did nt Understand.....Can u Explain...
Ishaan94
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\[|z| \geq 3\implies z\leq-3\;or\;z\geq3\]
\(\left|z + \frac1z\right|\) is just a symmetrical hyperbola about Y-axis. If you're well versed with hyperbola you will get the idea of the graph with constraint \(|z|\geq3\).
Yahoo!
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lol...i Dont knw Hyperbola....
Ishaan94
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Okay.
\[x + \frac1x\]
\(x\) is ever increasing, so to find the minima of \(x + \frac1x\) you should find value for which \(x\) is minimum i.e 3 as \(1/x\) is decreasing for \(x>1\).
Ishaan94
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Important to note \(1/x\) is decreasing not negative.
Ishaan94
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Negative with respect to \(x\).
Pallavi06
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express Z= x+ iy.......and its mod value is suar root of x^2 +y^2.....now mod of Z+1 can be expressed as (x+1 )+ iy......and then solve it.....i know its very lengthy but it works when u have no other option....
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If r is the modulus of z, then |z+ 1/z| >= r - 1/r.
Now the minimum of r - 1/r for r>=3 is clearly 3 - 1/3 = 8/3.
The minimum for |z+1/z| is attained if |z| = 3 and if z and 1/z have opposite directions.
Take z = 3i so that z = 1/3i = - i / 3, and the minimum 8/3 is attained for that value.
Can u Explain this!
Ishaan94
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I was afk.
Complex Numbers?
Yahoo!
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afk. ===?
Ishaan94
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away from the keyboard.
Vaidehi09
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yup. complex nos.
Yahoo!
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Yes.. it is Complex Numbers
Pallavi06
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let Z= x + iy and try to get value of mod 1/z
Pallavi06
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can u do it?
Pallavi06
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if mod z is r then ull get mod i/z as 1/r.....
Ishaan94
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I will be back, my trackpad isn't working somehow.
Vaidehi09
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shouldn't it be this way? or is something wrong with this?
|dw:1347108277591:dw|
hartnn
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if its <= then its max value , not least value.
Yahoo!
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The answer should be 8/3
hartnn
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and yes,thats the correct way to find max value.
Yahoo!
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lol.....Can we Do like this
|z -(- 1/z)| > = |z| - |-1/z|
Yahoo!
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= 3 - 1/3 = 8/3
hartnn
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is this true ?
|x-y| >= |x|-|y|
if yes, then u can do it like that.
hartnn
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and i guess its true.
Yahoo!
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Yes.....