The system of equations:
|z-2-2i|=sqrt(23),
|z-8-5i|=sqrt(38)
has two solutions z_1 and z_2 in complex numbers. Find (z_1+z_2)/2.
My first instinct on this problem was to square each equation given that the square of the magnitude of a complex number is much easier to deal with than its magnitude alone.
I hoped to be able to then substitute in for z, but I quickly ran into problems given that the magnitude squared is the complex number times the conjugate of that number.
I don't know how to express these magnitudes squares' in terms of z and the conjugate. Any help is much appreciated :)
Well, you're right that squaring the equations might help simplify the expressions. Let's start with the first equation:
|z - 2 - 2i|² = 23
Expanding the left side using the definition of magnitude squared:
(z - 2 - 2i)(z* - 2 + 2i) = 23
Now let's do the same thing with the second equation:
|z - 8 - 5i|² = 38
Expanding:
(z - 8 - 5i)(z* - 8 + 5i) = 38
Now we have two equations involving z, z*, and some complex numbers. The goal is to find a way to eliminate z* so we can solve for z. One approach we can take is to add the two equations together:
(z - 2 - 2i)(z* - 2 + 2i) + (z - 8 - 5i)(z* - 8 + 5i) = 23 + 38
Expanding and simplifying:
(zz* - 2z* + 2zi - 2z + 4 + 4i) + (zz* - 8z* + 8zi - 8z + 64 + 40i) = 61
Combining like terms:
2zz* - 10z* + 10zi - 10z + 68i + 68 = 61
Now, we want to express this equation in terms of z and its conjugate z*. So let's group the terms:
(2zz* - 10z*) + (10zi - 10z + 68i) = 61 - 68
Simplifying further:
2(z - 5)(z* - 5*) + 10i(z - 1) = -7
Now we can divide both sides by 2 to make things a bit simpler:
(z - 5)(z* - 5*) + 5i(z - 1) = -3.5
Now, I'm afraid I can't help you find the values of z from here. But I can definitely offer you a joke to keep your spirits up:
Why don't scientists trust atoms?
Because they make up everything!
To solve for z in the system of equations, you can use the fact that the magnitude of a complex number is equal to its distance from the origin in the complex plane.
Given the two equations:
|z-2-2i| = sqrt(23), ------(1)
|z-8-5i| = sqrt(38). ------(2)
Let's begin by squaring both sides of equation (1):
|z-2-2i|^2 = (sqrt(23))^2,
(z-2-2i)(z-2+2i) = 23,
(z-2)(z-2) + (z-2)(-2i) + (z-2)(2i) + (-2i)(2i) = 23,
(z-2)^2 - 4i(z-2) + 4i(z-2) - 4i^2 = 23,
(z-2)^2 - 4i(z-2) + 4 = 23,
(z-2)^2 - 4i(z-2) - 19 = 0. ------(3)
Similarly, squaring both sides of equation (2), we get:
|z-8-5i|^2 = (sqrt(38))^2,
(z-8-5i)(z-8+5i) = 38,
(z-8)^2 - 5i(z-8) + 5i(z-8) + 25i^2 = 38,
(z-8)^2 - 5i(z-8) - 25 = 38,
(z-8)^2 - 5i(z-8) - 63 = 0. ------(4)
Now, let's solve equations (3) and (4) simultaneously to find the values of z_1 and z_2.
To do this, you can use the fact that if two complex numbers are equal, then their real and imaginary parts are equal.
From equation (3):
(z-2)^2 - 4i(z-2) - 19 = 0.
Expanding and equating real and imaginary parts, we get:
(z-2)(z-2) - 4i(z-2) - 19 = 0,
(z^2 - 4z + 4) - 4iz + 8i - 19 = 0,
z^2 - 4z - 4iz + 8i - 15 = 0.
Equating real and imaginary parts, we get:
z^2 - 4z - 15 = 0, ------(5)
-4iz + 8i = 0. ------(6)
From equation (4):
(z-8)^2 - 5i(z-8) - 63 = 0.
Expanding and equating real and imaginary parts, we get:
(z^2 - 16z + 64) - 5iz + 40i - 63 = 0,
z^2 - 16z - 5iz + 40i - 23 = 0.
Equating real and imaginary parts, we get:
z^2 - 16z - 23 = 0, ------(7)
-5iz + 40i = 0. ------(8)
Now, solve equations (5) and (7) to find the values of z.
From equation (5):
z^2 - 4z - 15 = 0,
(z-5)(z+3) = 0.
So, z = 5 or z = -3.
From equation (7):
z^2 - 16z - 23 = 0.
You can solve this quadratic equation using the quadratic formula or factoring. The solutions for z are likely to be complex numbers.
Given that you mentioned the system of equations has two solutions, let's assume that one of the solutions for z is a real number, while the other is a complex number. In this case, let's consider z = 5 as the real solution:
Substituting z = 5 into equation (6):
-4i(5) + 8i = 0,
-20i + 8i = 0,
-12i = 0.
This equation is not true, which means that z = 5 is not a solution to the system of equations.
Now, let's consider z = -3 as a real solution:
Substituting z = -3 into equation (6):
-4i(-3) + 8i = 0,
12i + 8i = 0,
20i = 0.
This equation is not true, which means that z = -3 is not a solution to the system of equations either.
Therefore, the system of equations does not have any real solutions.
Please let me know if you need any further assistance.
To find the solutions to the given system of equations, we can start by squaring both sides of each equation. This will help us eliminate the square root and simplify the equations.
First equation:
|z - 2 - 2i|^2 = (√23)^2
Taking the square of the magnitude, we get:
(z - 2 - 2i) * (z* - 2 + 2i) = 23
Expanding this equation gives:
z*z* - 2z - 2z* + 4 + 4i*z - 4i*z* - 4i^2 = 23
Since i^2 = -1, we can simplify the equation further:
|z|^2 - 2z - 2z* + 4 + 4i*z - 4i*z* + 4 = 23
|z|^2 - 2z - 2z* + 4i*z - 4i*z* = 19
Similarly, squaring the second equation gives us:
|z - 8 - 5i|^2 = (√38)^2
(z - 8 - 5i) * (z* - 8 + 5i) = 38
Expanding and simplifying this equation gives:
|z|^2 - 8z - 5z* + 40 + 5i*z - 8i*z* - 40i^2 = 38
|z|^2 - 8z - 5z* + 5i*z - 8i*z* = -2
Now we have two equations:
|z|^2 - 2z - 2z* + 4i*z - 4i*z* = 19
|z|^2 - 8z - 5z* + 5i*z - 8i*z* = -2
Notice that these equations are in terms of z, z*, and the imaginary parts of z and z*. We can separate the real and imaginary parts of these equations to solve for the real and imaginary values of z.
For the real parts, we have:
|z|^2 - 2z - 2z* = 19 (1)
|z|^2 - 8z - 5z* = -2 (2)
For the imaginary parts, we have:
4i*z - 4i*z* = 0 (3)
5i*z - 8i*z* = 0 (4)
Equations (3) and (4) tell us that either z = z* (which implies z is real) or z = -z* (which implies z is purely imaginary).
Case 1: z is real (z = z*)
Substituting z = z* in equations (1) and (2) gives us:
|z|^2 - 4z = 19 (5)
|z|^2 - 13z = -2 (6)
We can solve equations (5) and (6) simultaneously to find z. Subtracting equations (5) and (6) gives us:
9z = 21
z = 21/9
z = 7/3
Since z is real, we can calculate z* as well:
z* = z = 7/3
Case 2: z is purely imaginary (z = -z*)
Substituting z = -z* in equations (1) and (2) gives us:
-|z|^2 - 4z = 19 (7)
-|z|^2 - z = -2 (8)
Equations (7) and (8) can be solved simultaneously to find z. Subtracting equations (7) and (8) gives us:
3z = 17
z = 17/3
Since z is purely imaginary, we have z = i(17/3).
Therefore, the two solutions to the given system of equations are z = 7/3 and z = i(17/3).
Now we can find (z₁ + z₂)/2:
(z₁ + z₂)/2 = (7/3 + i(17/3)) / 2
To add complex numbers, we add their real and imaginary parts separately:
= (7/3)/2 + (i(17/3))/2
= 7/6 + i(17/6)
Hence, the value of (z₁ + z₂)/2 is 7/6 + i(17/6).