Let  z
and  w
be complex numbers that satisfy  z^2+zw*=50
and  2z*=w*(1−i)
with  Re(z)>0
. Find  w

To find the complex number \( w \) given the conditions:

1. \( z^2 + zw^* = 50 \)
2. \( 2z^* = w^*(1 - i) \)
3. \( \text{Re}(z) > 0 \)

we will follow a step-by-step approach to solve for \( w \).

### Step 1: Analyze the Equations

Given the complex numbers \( z \) and \( w \), we denote them as:
\[
z = x + yi, \quad w = u + vi
\]
where \( x, y, u, \) and \( v \) are real numbers.

The conjugates \( z^* \) and \( w^* \) are:
\[
z^* = x - yi, \quad w^* = u - vi
\]

Substitute these into the given equations.

### Step 2: Substitute in the Second Equation

The second equation is:
\[
2z^* = w^*(1 - i)
\]
Substitute the conjugates:
\[
2(x - yi) = (u - vi)(1 - i)
\]

Expand and equate real and imaginary parts:
\[
2x - 2yi = (u - vi)(1 - i) = (u - v - (u + v)i)
\]

Equating real and imaginary parts:
\[
2x = u - v, \quad -2y = -u - v
\]

Rewriting these equations, we get:
\[
u = x + y, \quad v = x - y
\]

### Step 3: Substitute in the First Equation

Now use these in the first equation:
\[
z^2 + zw^* = 50
\]

Compute \( z^2 \) and \( zw^* \):
\[
z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi
\]
\[
zw^* = (x + yi)(u - vi) = (x + yi)((x + y) - (x - y)i)
\]

Expand \( zw^* \):
\[
zw^* = x(x + y) + y(x - y) + i(y(x + y) - x(x - y))
\]

Simplify \( zw^* \):
\[
zw^* = (x^2 + xy + yx - y^2) + i(yx + y^2 - x^2 + xy)
\]
\[
zw^* = (x^2 - y^2 + 2xy) + i(-x^2 + y^2 + 2xy)
\]

Thus, the equation \( z^2 + zw^* = 50 \) becomes:
\[
(x^2 - y^2 + 2xy) + (x^2 - y^2 + 2xy) = 50
\]
\[
2(x^2 - y^2 + 2xy) = 50
\]

Simplify:
\[
x^2 - y^2 + 2xy = 25
\]

### Step 4: Solve for \( z \) and \( w \)

Now use the real numbers:
\[
x^2 + y^2 = r^2, \quad xy = s
\]
where \( r^2 = x^2 + y^2 \) and \( s = xy \).

Thus, \( x^2 - y^2 + 2xy = 25 \), substitute:
\[
r^2 + 2s = 25
\]
and for the previous solution \( r = 5 \), thus \( x = y = \frac{5}{\sqrt{2}} \).

Now find \( w \):
\[
w = x + y + i(x - y) = 5 + i \times 0
\]

Thus, \( w = 5 \).

### Solution:

The value of \( w \) is:
\[
\boxed{5}
\]

Explore the solution to complex equations involving complex numbers z and w. Understand how to solve equations like z^2 + zw* = 50 and 2z* = w*(1 - i) with real part of z greater than 0. Suitable for advanced mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation