## anonymous 3 years ago $|Z+ \frac{ 9 }{ Z }| = 6$ ,Then find the Greatest value of |Z| ?

1. JamesJ

Easiest way to proceed is the square both sides and see what you get from there

2. JamesJ

so ... waiting for you to take a step or engage in the conversation

3. anonymous

couldnt you also just say Z + 9/Z = 6, and ignore the absolute value, because you are looking for the largest value anyways then solve for Z as: Z^2 + 9 = 6z Z^2 - 6x + 9 = 0 and go from there

4. JamesJ

that gives you some of the solutions. You also need to consider the case z + 9/z = -6

5. anonymous

@JamesJ that would give answer as |Z| = 3.... But the answer shuld be |Z| = 3 + sqrt18

6. anonymous

Note: Z is a Complex Number

7. JamesJ

Squaring both sides $\left| z + 9/z \right|^2 = 6^2$ i.e., $(z + 9/z)^2 = 36$ as |x|^2 = x^2 for all real numbers x. Hence $z^2 + 18 + 81/z^2 = 36$ and therefore $z^4 - 18z^2 + 81 = 0$ Now you have a quadratic equation in z^2 which you can solve... $(z^2 - 9)^2 = 0$ $z^2 = 9$ $|z| = 3$ yes. The other answer you just wrote down doesn't satisfy the original equation, does it?

8. JamesJ

Or is z here a complex number?

9. anonymous

Z is a Complex Number

10. JamesJ

If z is complex, we're going to have use a different approach.

11. JamesJ

right. So in that case, what have you tried?

12. anonymous

No Idea..

13. JamesJ

Write z = x +iy as usual. Now notice the original equation is equivalent to |z^2 + 9| = 6|z| Now write out those two expressions in terms of x and y and see what you get. It will be easier to square them.

14. JamesJ

i.e., write out $|z^2 + 9|^2 = 36 |z|^2$ for example, $|z|^2 = x^2 + y^2$ You take the next step ....

15. anonymous

(x^2 -y^2 + 9)^2 + (2xy)^2 = 36(x^2 + y^2)

16. JamesJ

Yes, now see if you can reduce this somehow

17. experimentX

|dw:1360344301541:dw|

18. JamesJ

Yes, it's not clear to me that we get much from the polar approach. Where do you go from there? Have you studied Lagrange multipliers btw? This problem looks a lot like a Lagrangian problem. You have a constraint in x and y given by this equation, and you want to maximize f(x,y) = x^2 + y^2

19. anonymous

Nope..i havent Studied that

20. experimentX

|dw:1360344512672:dw|

21. experimentX

|dw:1360344710864:dw| damn this is getting complicated.

22. experimentX

|dw:1360344933595:dw|

23. sirm3d

$6=\left|Z+\frac{9}{Z}\right|>\left||Z|-\left|\frac{9}{Z}\right|\right|$put $z=re^{i\theta}\\|z|=r,\quad \left|\frac{9}{z}\right|=\frac{1}{r}$ now solve the inequality in r

24. experimentX

since we have isolated the value r as function of theta we might maximize it using simple derivatives. but this is not really suggested.

25. sirm3d

oops, that's 9/r |r-9/r|<6 r-9/r < 6 r^2 - 9 < 6r r^2 - 6r < 9 r^2 - 6r + 9 < 18 (r-3)^2 < 18 (r-3) < sqrt(18) r< 3 + sqrt(18)

26. JamesJ

very nice, yes.