﻿Sketch the set of points determined by the given equations. Be sure the draw the coordinate axis
(
real and imaginary
)
 ﻿as well as indicate the coordinates of several points on the sets to give a sense of scale.
(
a
)
 
|
z 
−
 ﻿i 
+
 
3
|
 
=
 
1
(
b
)
 
|
2
z 
−
 
3
i 
+
 
1
|
 
<
=
 
2
(
c
)
 ﻿z
^
2
 
+
 ﻿z
^
2
 
=
 
2
.
 ﻿Additionally, verify that this is a hyperbola. The second z
^
2
 ﻿in the equation has a line on top like this 
-
step by step solution with summaries answer

Let's address each part of the question step-by-step, including graph sketches and analyses of the complex numbers, geometry, and verifying if the given curve is a hyperbola.

### Part (a) \(| z - i + 3 | = 1\)
This equation is in the form \(| z - c | = r\), which represents a circle in the complex plane. Here, \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part of the complex number.

**Solution steps:**
1. The expression can be rewritten as \(|z - (-3 + i)| = 1\).
2. This describes a circle with a center at \((-3, 1)\) in the complex plane (real part \(-3\), imaginary part \(1\)).
3. The radius of the circle is \(1\).

**Summary for (a):**
- The set of points is a circle centered at \((-3, 1)\) with a radius of \(1\).
- This will be sketched as a circle in the complex plane, with several key points marked.

### Part (b) \(| 2z - 3i + 1 | \leq 2\)
Here we analyze the inequality.

**Solution steps:**
1. Rewrite the equation in terms of \(z = x + yi\):
   \[
   |2(x + yi) - 3i + 1| \leq 2
   \]
   Simplify this:
   \[
   |2x + (2y - 3)i + 1| \leq 2
   \]
2. This inequality describes a disk centered at \((-1/2, 3/2)\) (real part \(-\frac{1}{2}\), imaginary part \(\frac{3}{2}\)) with a radius of \(1\).
3. The points satisfying this inequality form a closed disk in the complex plane.

**Summary for (b):**
- The set of points is a disk centered at \((-1/2, 3/2)\) with a radius of \(1\), including the boundary of the disk.
- This will be sketched as a filled circle.

### Part (c) \(z^2 + \overline{z^2} = 2\)

This equation involves both \(z^2\) and its complex conjugate \(\overline{z^2}\).

**Solution steps:**
1. Let \(z = x + yi\), where \(x\) and \(y\) are real and imaginary parts of \(z\), respectively.
2. Compute \(z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi\), and \(\overline{z^2} = (x^2 - y^2 - 2xyi)\).
3. Add \(z^2\) and \(\overline{z^2}\):
   \[
   z^2 + \overline{z^2} = (x^2 - y^2 + 2xyi) + (x^2 - y^2 - 2xyi) = 2(x^2 - y^2)
   \]
4. Set this equal to 2:
   \[
   2(x^2 - y^2) = 2
   \]
   Dividing both sides by 2:
   \[
   x^2 - y^2 = 1
   \]
5. This is the standard equation of a hyperbola.

**Summary for (c):**
- The equation \(z^2 + \overline{z^2} = 2\) represents a hyperbola with the equation \(x^2 - y^2 = 1\).
- The graph will be a hyperbola in the complex plane.

---

### Verifying Hyperbola:
We derived \(x^2 - y^2 = 1\), which is the equation of a hyperbola, confirming that the equation represents a hyperbola in the complex plane.

### Sketching the Sets:
1. For part (a), a circle centered at \((-3, 1)\) with radius 1.
2. For part (b), a disk centered at \((-1/2, 3/2)\) with radius 1.
3. For part (c), a hyperbola given by \(x^2 - y^2 = 1\), symmetric about the x-axis.

Would you like to see the sketch of the graphs?

---

### 5 Related Questions:
1. How do you convert a complex number into polar form?
2. How do you derive the equation of a circle in the complex plane?
3. What is the significance of the magnitude of a complex number in geometry?
4. Can complex conjugates be used to simplify equations involving imaginary parts?
5. What are the geometric interpretations of inequalities involving complex numbers?

### Tip:
To verify if a given curve is a hyperbola or another conic section, look for terms like \(x^2 - y^2\) in the equation.

Sketch the set of points determined by the given equations. Be sure the draw the coordinate axis (real and imaginary) as well as indicate the coordinates of several points on the sets to give a sense of scale. (a) |z - i + 3| = 1 (b) |2z - 3i + 1| <= 2 (c) z^2 + z^2 = 2. Additionally, verify that this is a hyperbola.

Explore how to graph complex numbers with equations like |z - i + 3| = 1 and |2z - 3i + 1| <= 2. Learn how to verify that the equation z^2 + z^2 = 2 represents a hyperbola. This step-by-step solution covers complex plane geometry, inequalities, and hyperbolas. Suitable for advanced high school and undergraduate math students.

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2 . Additionally, verify that this is a hyperbola. The second z ^ 2 in the equation has a line on top like this

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