For z=x+iy𝑧=𝑥+𝑖𝑦 find the values x𝑥 and y𝑦 satisfying 2z1+i−2zi=52+i2𝑧1+𝑖−2𝑧𝑖=52+𝑖.

1. Break Down the Problem

We need to solve the equation: $2z \frac{1+i-2zi}{52+i2z} = 1$ where $z = x + iy$. This requires simplifying both sides and isolating the variables $x$ and $y$.

2. Relevant Concepts

1. Substitute $z = x + iy$.
2. Use properties of complex numbers to simplify the equation.
3. Set the real and imaginary parts equal to isolate the variables.

3. Analysis and Detail

Step 1: Substitute $z$

Substituting $z = x + iy$ into the equation:

$2(x + iy) \frac{1+i-2i(x+iy)}{52+i2(x+iy)} = 1$

This simplifies to:

Step 2: Simplifying the numerator

$1 + i - 2i(x + iy) = 1 + i - 2ix + 2y = (1 + 2y) + i(1 - 2x)$

So, the numerator becomes: $2(x + iy) \left( (1 + 2y) + i(1 - 2x) \right)$

This gives: $2x(1 + 2y) + 2iy(1 + 2y) + 2i(1 - 2x) + 2y(1 - 2x)$

Step 3: Simplifying the denominator

For the denominator: $52 + i2(x + iy) = 52 + 2ix - 2y$ So, it becomes: $52 + 2ix - 2y$

Step 4: Set up the equation

Now, rewrite the equation in simpler form: $\frac{2x(1 + 2y) + 2i(1 - 2x)}{52 - 2y + 2ix} = 1$

Step 5: Cross-multiply

Cross-multiplying gives: $2x(1 + 2y) + 2i(1 - 2x) = 52 - 2y + 2ix$

Step 6: Equate real and imaginary parts

We can now separate real and imaginary components:

1. Real part: $2x(1 + 2y) = 52 - 2y$
2. Imaginary part: $2(1 - 2x) = 2x$

4. Verify and Summarize

Solving the imaginary part: $2 - 4x = 2x \implies 2 = 6x \implies x = \frac{1}{3}$

Now substitute $x = \frac{1}{3}$ back into the real part: $2 \cdot \frac{1}{3}(1 + 2y) = 52 - 2y$ This simplifies to: $\frac{2}{3}(1 + 2y) = 52 - 2y$ Multiply through by 3: $2(1 + 2y) = 156 - 6y \implies 2 + 4y = 156 - 6y$ Combine terms: $10y = 154 \implies y = \frac{154}{10} = 15.4$

Final Answer

Thus, the values of $x$ and $y$ that satisfy the equation are: $x = \frac{1}{3}, \quad y = 15.4$