Question

\[|Z+ \frac{ 9 }{ Z }| = 6\] ,Then find the Greatest value of |Z| ?

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

Note: Z is a Complex Number

Answer 7

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?

Answer 8

Or is z here a complex number?

Answer 9

Z is a Complex Number

Answer 10

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

Answer 11

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

Answer 12

No Idea..

Answer 13

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.

Answer 14

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

Answer 15

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

Answer 16

Yes, now see if you can reduce this somehow