Yahoo! 3 years ago if |z|>= 3 the least value of |z + 1/z| is?

1. hartnn

i think we need to use |x|+|y|>|x+y| but not sure.

2. Yahoo!

But wat Does that Least Value For?

3. hartnn

i m just trying.... if |z|>=3 what can u say about |1/z| ??

4. Yahoo!

|z -(- 1/z)| > = |z| - |-1/z|

5. Yahoo!

I Did nt Understand....wat is here Least..""

6. hartnn

means |z+1/z| is > = 'some number' u need to find that number.

7. hartnn

but |z+1/z| <= |z|+|1/z| how did u get > sign there ?

8. Yahoo!

u see..i have inserted a -ve there....Since it asks for least value."

9. Callisto

Hmm.. Am I right for this: |z| ≥ 3 i.e. z ≥ 3 or z ≤ -3 ?

10. hartnn

i think z is complex here, isn't it @Yahoo! ??

11. Yahoo!

Yes!

12. hartnn

so |z|>= 3 represents region outside the circle with radius 3.....if i have not mis-interpreted

13. Callisto

Wow! Something I don't know! I'm sorry!!

14. Yahoo!

"OUTSIDE" hw

15. hartnn

because of > sign, if < then inside.

16. Yahoo!

Ok...)

17. Yahoo!

@hartnn Any Idea....

18. hartnn

i got something from yahoo answers , trying to understand it, see if that makes some sense to u..... http://in.answers.yahoo.com/question/index?qid=20100423221137AAAgV9K

19. Yahoo!

i think we have to find the least dist b/w Circles

20. Yahoo!

@hartnn i Did nt Understand.....Can u Explain...

21. Ishaan94

$|z| \geq 3\implies z\leq-3\;or\;z\geq3$ $$\left|z + \frac1z\right|$$ 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 $$|z|\geq3$$.

22. Yahoo!

lol...i Dont knw Hyperbola....

23. Ishaan94

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 $$x>1$$.

24. Ishaan94

Important to note $$1/x$$ is decreasing not negative.

25. Ishaan94

Negative with respect to $$x$$.

26. Pallavi06

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....

27. Yahoo!

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!

28. Ishaan94

I was afk. Complex Numbers?

29. Yahoo!

afk. ===?

30. Ishaan94

away from the keyboard.

31. Vaidehi09

yup. complex nos.

32. Yahoo!

Yes.. it is Complex Numbers

33. Pallavi06

let Z= x + iy and try to get value of mod 1/z

34. Pallavi06

can u do it?

35. Pallavi06

if mod z is r then ull get mod i/z as 1/r.....

36. Ishaan94

I will be back, my trackpad isn't working somehow.

37. Vaidehi09

shouldn't it be this way? or is something wrong with this? |dw:1347108277591:dw|

38. hartnn

if its <= then its max value , not least value.

39. Yahoo!

40. hartnn

and yes,thats the correct way to find max value.

41. Yahoo!

lol.....Can we Do like this |z -(- 1/z)| > = |z| - |-1/z|

42. Yahoo!

= 3 - 1/3 = 8/3

43. hartnn

is this true ? |x-y| >= |x|-|y| if yes, then u can do it like that.

44. hartnn

and i guess its true.

45. Yahoo!

Yes.....