he complex number z1,z2𝑧1,𝑧2 satisfies the equation z+1+8 i=|z|(1+i)𝑧+1+8 𝑖=𝑧1+𝑖, where i=−1−−−√𝑖=-1 then the equation whose roots are |z1|𝑧1 and |z2|𝑧2 is

To find the equation whose roots are given in the form of $|z_1| z_1$ and $|z_2| z_2$ based on the initial equation provided, we'll follow the required steps to solve the complex equation.

1. Break Down the Problem

We start with the equation provided: $z + 1 + 8i = |z|(1+i)$ and we need to express this in terms of specific roots $z_1$ and $z_2$.

2. Relevant Concepts

We recognize that $|z|$ denotes the modulus of the complex number $z$. The term $1 + i$ can be simplified, and the question is likely looking for a polynomial with roots related to these complex numbers.

Let’s express $z$ in its standard form: $z = x + yi$ where $x$ and $y$ are real numbers.

3. Analysis and Detail

We now plug $z$ into the equation: $(x + yi) + 1 + 8i = \sqrt{x^2 + y^2}(1+i)$ Simplifying the left side gives: $(x + 1) + (y + 8)i$

Now equate the real and imaginary parts:

Real part: $x + 1 = \sqrt{x^2 + y^2}$

Imaginary part: $y + 8 = \sqrt{x^2 + y^2}$

Now we have two equations:

1. $x + 1 = \sqrt{x^2 + y^2}$ (1)
2. $y + 8 = \sqrt{x^2 + y^2}$ (2)

Setting (1) equal to (2): $x + 1 = y + 8$ This provides us an expression for $y$: $y = x - 7$

Substituting $y$ back into either equation (let's use (1)): $x + 1 = \sqrt{x^2 + (x - 7)^2}$ Squaring both sides: $(x + 1)^2 = x^2 + (x - 7)^2$ Expanding gives: $x^2 + 2x + 1 = x^2 + x^2 - 14x + 49$ $2x + 1 = x^2 - 14x + 49$ Reorganizing yields: $0 = x^2 - 16x + 48$

4. Verify and Summarize

We solve this quadratic equation using the $x$: $x = \frac{16 \pm \sqrt{(16)^2 - 4 \cdot 1 \cdot 48}}{2 \cdot 1}$ $x = \frac{16 \pm \sqrt{256 - 192}}{2} = \frac{16 \pm \sqrt{64}}{2} = \frac{16 \pm 8}{2}$

Thus: $x = 12 \quad \text{or} \quad x = 4$ Now substitute back to get $y$:

• For $x = 12$: $y = 12 - 7 = 5$ → $z_1 = 12 + 5i$
• For $x = 4$: $y = 4 - 7 = -3$ → $z_2 = 4 - 3i$

The moduli are: $|z_1| = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$ $|z_2| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Final Answer

The roots of the desired equation are $|z_1| z_1 = 13(12 + 5i)$ and $|z_2| z_2 = 5(4 - 3i)$. Thus, the equation whose roots are given by these complex numbers can be formulated as: $x^2 - (|z_1| + |z_2|)x + |z_1||z_2| = 0$ with roots $156 + 65i$ and $20 - 15i$.