Let z and w be complex numbers such that
|2z-2|=25,
|z+2w|=5, and
|z+w|=2.
Find |z|.
I first squared both equations and expressed the square of the magnitudes in terms of the complex numbers and its conjugates. Next, I summed the equations together and attempted to solve for |z|.
My equation got very messy very quickly and if there is an easier way to go about doing this. Any help would be very much appreciated!
To find the value of |z|, here is an alternative approach that might simplify the process:
Let's start by assigning variables to the magnitudes of z and w. Let |z| = a and |w| = b.
Now, let's substitute these variables into the given equations:
|2z-2| = 25 ---> |2(a-1)| = 25 ---> |2a-2| = 25 (equation 1)
|z+2w| = 5 ---> |a+2b| = 5 (equation 2)
|z+w| = 2 ---> |a+b| = 2 (equation 3)
We can now work with these simplified equations to find the value of |z|.
First, let's square equation 3:
|a+b|^2 = (2)^2
(a+b)(a+b*) = 4
a^2 + ab* + ba + b^2 = 4
a^2 + 2ab* + b^2 = 4
Next, let's square equation 2:
|a+2b|^2 = 5^2
(a+2b)(a+2b*) = 25
a^2 + 2ab* + 4b^2 = 25
Now, let's subtract the squared version of equation 3 from the squared version of equation 2:
(a^2 + 2ab* + 4b^2) - (a^2 + 2ab* + b^2) = 25 - 4
3b^2 = 21
b^2 = 7
Now, let's substitute the value of b^2 into equation 3:
a^2 + 2ab* + 7 = 4
a^2 + 2ab* = -3
We can now substitute the value of a^2 + 2ab* into equation 1:
|2a-2|^2 = 25
(2a-2)(2a-2*) = 25
4(a^2 - ab* - a* b + b^2 - a + a*) = 25
4(a^2 - 2ab* + b^2 - a* b - a + a*) = 25
4(a^2 - 2ab* + b^2 - 2ab) = 25
4(a^2 - 4ab + b^2) = 25
4((a-b)^2) = 25
(a-b)^2 = 25/4
(a-b) = ± √(25/4)
a-b = ± 5/2
Now, let's explore the two possible values for a-b:
Case 1: a-b = 5/2
If a-b = 5/2, we can solve equation 3 for a+b:
a+b = 2
Now, let's solve the system of equations with a-b = 5/2 and a+b = 2:
Adding the equations:
2a = 9/2
a = 9/4
Substituting a = 9/4 into equation 3 to solve for b:
(9/4) + b = 2
b = 8/4 - 9/4
b = -1/4
Thus, one possible solution is a = 9/4 and b = -1/4.
Case 2: a-b = -5/2
If a-b = -5/2, we can solve equation 3 for a+b:
a+b = 2
Now, let's solve the system of equations with a-b = -5/2 and a+b = 2:
Adding the equations:
2a = 3/2
a = 3/4
Substituting a = 3/4 into equation 3 to solve for b:
(3/4) + b = 2
b = 8/4 - 3/4
b = 5/4
Thus, another possible solution is a = 3/4 and b = 5/4.
In summary, there are two possible values for |z|:
1. |z| = a = 9/4
2. |z| = a = 3/4
Therefore, |z| can be either 9/4 or 3/4.