1. x_{2} ^ 2 + y_{2} ^ 2 - 16x + 6y - 48 = 0

2. x ^ 2 + y ^ 2 + 14x + 8y + 1 = 0

3. x ^ 2 + y ^ 2 + 4x - 10y + 4 = 0

4. x ^ 2 + y ^ 2 + 18x - 8y + 61 = 0

These equations are of the form \( x^2 + y^2 + Dx + Ey + F = 0 \), which can be converted into the standard form of a circle equation \( (x - h)^2 + (y - k)^2 = r^2 \) by completing the square. Let's go through each one step by step.

### 1. \( x_{2}^2 + y_{2}^2 - 16x + 6y - 48 = 0 \)

#### Complete the square:
- For \( x \): 
  \[
  x^2 - 16x \rightarrow (x - 8)^2 - 64
  \]
- For \( y \): 
  \[
  y^2 + 6y \rightarrow (y + 3)^2 - 9
  \]

Substitute into the equation:
\[
(x - 8)^2 - 64 + (y + 3)^2 - 9 - 48 = 0
\]
Combine constants:
\[
(x - 8)^2 + (y + 3)^2 - 121 = 0
\]
\[
(x - 8)^2 + (y + 3)^2 = 121
\]
So, the equation represents a circle with center \((h, k) = (8, -3)\) and radius \( r = 11 \).

### 2. \( x^2 + y^2 + 14x + 8y + 1 = 0 \)

#### Complete the square:
- For \( x \): 
  \[
  x^2 + 14x \rightarrow (x + 7)^2 - 49
  \]
- For \( y \): 
  \[
  y^2 + 8y \rightarrow (y + 4)^2 - 16
  \]

Substitute into the equation:
\[
(x + 7)^2 - 49 + (y + 4)^2 - 16 + 1 = 0
\]
Combine constants:
\[
(x + 7)^2 + (y + 4)^2 - 64 = 0
\]
\[
(x + 7)^2 + (y + 4)^2 = 64
\]
So, the equation represents a circle with center \((h, k) = (-7, -4)\) and radius \( r = 8 \).

### 3. \( x^2 + y^2 + 4x - 10y + 4 = 0 \)

#### Complete the square:
- For \( x \): 
  \[
  x^2 + 4x \rightarrow (x + 2)^2 - 4
  \]
- For \( y \): 
  \[
  y^2 - 10y \rightarrow (y - 5)^2 - 25
  \]

Substitute into the equation:
\[
(x + 2)^2 - 4 + (y - 5)^2 - 25 + 4 = 0
\]
Combine constants:
\[
(x + 2)^2 + (y - 5)^2 - 25 = 0
\]
\[
(x + 2)^2 + (y - 5)^2 = 25
\]
So, the equation represents a circle with center \((h, k) = (-2, 5)\) and radius \( r = 5 \).

### 4. \( x^2 + y^2 + 18x - 8y + 61 = 0 \)

#### Complete the square:
- For \( x \): 
  \[
  x^2 + 18x \rightarrow (x + 9)^2 - 81
  \]
- For \( y \): 
  \[
  y^2 - 8y \rightarrow (y - 4)^2 - 16
  \]

Substitute into the equation:
\[
(x + 9)^2 - 81 + (y - 4)^2 - 16 + 61 = 0
\]
Combine constants:
\[
(x + 9)^2 + (y - 4)^2 - 36 = 0
\]
\[
(x + 9)^2 + (y - 4)^2 = 36
\]
So, the equation represents a circle with center \((h, k) = (-9, 4)\) and radius \( r = 6 \).

---

### Summary of Results:
1. **Equation**: \( (x - 8)^2 + (y + 3)^2 = 121 \)  
   **Center**: \((8, -3)\), **Radius**: \(11\)
2. **Equation**: \( (x + 7)^2 + (y + 4)^2 = 64 \)  
   **Center**: \((-7, -4)\), **Radius**: \(8\)
3. **Equation**: \( (x + 2)^2 + (y - 5)^2 = 25 \)  
   **Center**: \((-2, 5)\), **Radius**: \(5\)
4. **Equation**: \( (x + 9)^2 + (y - 4)^2 = 36 \)  
   **Center**: \((-9, 4)\), **Radius**: \(6\)

Would you like any more details on these steps?

### Follow-up Questions:
1. How do you determine the center and radius of a circle from a general equation?
2. What is the significance of completing the square in geometry?
3. Can these methods be applied to equations that represent ellipses or hyperbolas?
4. How would the method differ if the coefficients of \(x^2\) and \(y^2\) were not equal?
5. Could these circles ever intersect? If so, how would you find the points of intersection?

### Tip:
When working with quadratic equations representing circles, always start by rearranging the equation and completing the square to easily identify the circle's geometric properties.

Learn how to convert circle equations into the standard form (x - h)^2 + (y - k)^2 = r^2 by completing the square. Step-by-step solutions provided for circle equations involving completing the square method. Suitable for high school students and geometry enthusiasts.

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