Math worksheet for writing equations of circles in standard and general forms.

Problem Overview:

The task involves writing the standard form and general form equations of circles given their centers and radii. Let's break this down step by step.

#### Standard Form of a Circle
The standard form of the equation of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where:
- \((h, k)\) is the center of the circle,
- \(r\) is the radius of the circle.

#### General Form of a Circle
The general form of the equation of a circle is:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
This form can be derived from the standard form by expanding and rearranging terms.

---

Part 1: Standard Form Equations

We are given the center \((h, k)\) and radius \(r\) for each circle. Using the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can write the equations.

#### Problem 1:
- Center: \((12, -10)\)
- Radius: \(3\)

Substitute into the standard form:
\[
(x - 12)^2 + (y + 10)^2 = 3^2
\]
\[
(x - 12)^2 + (y + 10)^2 = 9
\]

#### Problem 2:
- Center: \((6, 11)\)
- Radius: \(5\)

Substitute into the standard form:
\[
(x - 6)^2 + (y - 11)^2 = 5^2
\]
\[
(x - 6)^2 + (y - 11)^2 = 25
\]

#### Problem 3:
- Center: \((-3, -2)\)
- Radius: \(6\)

Substitute into the standard form:
\[
(x + 3)^2 + (y + 2)^2 = 6^2
\]
\[
(x + 3)^2 + (y + 2)^2 = 36
\]

#### Problem 4:
- Center: \((-3, 8)\)
- Radius: \(3\)

Substitute into the standard form:
\[
(x + 3)^2 + (y - 8)^2 = 3^2
\]
\[
(x + 3)^2 + (y - 8)^2 = 9
\]

---

Part 2: General Form Equations

We are given the center \((h, k)\) and radius \(r\) for each circle. First, write the standard form, then expand it to get the general form.

#### Problem 5:
- Center: \((10, -16)\)
- Radius: \(\sqrt{3}\)

Standard Form:
\[
(x - 10)^2 + (y + 16)^2 = (\sqrt{3})^2
\]
\[
(x - 10)^2 + (y + 16)^2 = 3
\]

Expand to General Form:
\[
(x - 10)^2 = x^2 - 20x + 100
\]
\[
(y + 16)^2 = y^2 + 32y + 256
\]
\[
x^2 - 20x + 100 + y^2 + 32y + 256 = 3
\]
\[
x^2 + y^2 - 20x + 32y + 356 = 3
\]
\[
x^2 + y^2 - 20x + 32y + 353 = 0
\]

#### Problem 6:
- Center: \((-15, 2)\)
- Radius: \(2\)

Standard Form:
\[
(x + 15)^2 + (y - 2)^2 = 2^2
\]
\[
(x + 15)^2 + (y - 2)^2 = 4
\]

Expand to General Form:
\[
(x + 15)^2 = x^2 + 30x + 225
\]
\[
(y - 2)^2 = y^2 - 4y + 4
\]
\[
x^2 + 30x + 225 + y^2 - 4y + 4 = 4
\]
\[
x^2 + y^2 + 30x - 4y + 229 = 4
\]
\[
x^2 + y^2 + 30x - 4y + 225 = 0
\]

#### Problem 7:
- Center: \((16, 14)\)
- Radius: \(1\)

Standard Form:
\[
(x - 16)^2 + (y - 14)^2 = 1^2
\]
\[
(x - 16)^2 + (y - 14)^2 = 1
\]

Expand to General Form:
\[
(x - 16)^2 = x^2 - 32x + 256
\]
\[
(y - 14)^2 = y^2 - 28y + 196
\]
\[
x^2 - 32x + 256 + y^2 - 28y + 196 = 1
\]
\[
x^2 + y^2 - 32x - 28y + 452 = 1
\]
\[
x^2 + y^2 - 32x - 28y + 451 = 0
\]

#### Problem 8:
- Center: \((11, 5)\)
- Radius: \(4\)

Standard Form:
\[
(x - 11)^2 + (y - 5)^2 = 4^2
\]
\[
(x - 11)^2 + (y - 5)^2 = 16
\]

Expand to General Form:
\[
(x - 11)^2 = x^2 - 22x + 121
\]
\[
(y - 5)^2 = y^2 - 10y + 25
\]
\[
x^2 - 22x + 121 + y^2 - 10y + 25 = 16
\]
\[
x^2 + y^2 - 22x - 10y + 146 = 16
\]
\[
x^2 + y^2 - 22x - 10y + 130 = 0
\]

---

Part 3: Converting Given Equations to Standard Form

We are given equations in the form \((x - h)^2 + (y - k)^2 = r^2\) and need to identify the center \((h, k)\) and radius \(r\).

#### Problem 9:
\[
(x + 10)^2 + (y - 7)^2 = 25
\]
- Center: \((-10, 7)\)
- Radius: \(\sqrt{25} = 5\)

#### Problem 10:
\[
(x - 7)^2 + (y - 14)^2 = 19
\]
- Center: \((7, 14)\)
- Radius: \(\sqrt{19}\)

---

Final Answers:

#### Standard Form Equations:
1. \((x - 12)^2 + (y + 10)^2 = 9\)
2. \((x - 6)^2 + (y - 11)^2 = 25\)
3. \((x + 3)^2 + (y + 2)^2 = 36\)
4. \((x + 3)^2 + (y - 8)^2 = 9\)

#### General Form Equations:
5. \(x^2 + y^2 - 20x + 32y + 353 = 0\)
6. \(x^2 + y^2 + 30x - 4y + 225 = 0\)
7. \(x^2 + y^2 - 32x - 28y + 451 = 0\)
8. \(x^2 + y^2 - 22x - 10y + 130 = 0\)

#### Identified Centers and Radii:
9. Center: \((-10, 7)\), Radius: \(5\)
10. Center: \((7, 14)\), Radius: \(\sqrt{19}\)

\boxed{
\begin{aligned}
&\text{1. } (x - 12)^2 + (y + 10)^2 = 9 \\
&\text{2. } (x - 6)^2 + (y - 11)^2 = 25 \\
&\text{3. } (x + 3)^2 + (y + 2)^2 = 36 \\
&\text{4. } (x + 3)^2 + (y - 8)^2 = 9 \\
&\text{5. } x^2 + y^2 - 20x + 32y + 353 = 0 \\
&\text{6. } x^2 + y^2 + 30x - 4y + 225 = 0 \\
&\text{7. } x^2 + y^2 - 32x - 28y + 451 = 0 \\
&\text{8. } x^2 + y^2 - 22x - 10y + 130 = 0 \\
&\text{9. Center: } (-10, 7), \text{ Radius: } 5 \\
&\text{10. Center: } (7, 14), \text{ Radius: } \sqrt{19}
\end{aligned}
}