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Question 1
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Which answer choice shows the center of the circle with the equation x2 + y2 -8x +14y +57.

Question 1Select one:

a.
(4, -7)


b.
(7, 4)


c.
(-7, 4)


d.
(-4, 7)


Question 2
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What is the standard form of the equation of the circle x2 + y2 + 10x - 4y - 7 = 0?

Question 2Select one:

a.
(x + 5)2 + (y - 2)2 = 62


b.
(x + 5)2 + (y + 2)2 = 62


c.
(x - 5)2 + (y - 2)2 = 62


d.
(x - 5)2 + (y + 2)2 = 62

Let's solve each question step by step.

### Question 1: Finding the Center of the Circle

The equation of the circle is:

\[
x^2 + y^2 - 8x + 14y + 57 = 0
\]

To find the center of the circle, we need to rewrite the equation in the standard form:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle. We do this by completing the square for both \(x\) and \(y\).

#### Step-by-Step Solution:

1. **Group the \(x\) and \(y\) terms:**

\[
(x^2 - 8x) + (y^2 + 14y) = -57
\]

2. **Complete the square for \(x\):**

\[
x^2 - 8x \rightarrow (x - 4)^2 - 16
\]

3. **Complete the square for \(y\):**

\[
y^2 + 14y \rightarrow (y + 7)^2 - 49
\]

4. **Substitute back into the equation:**

\[
(x - 4)^2 - 16 + (y + 7)^2 - 49 = -57
\]

5. **Combine constants:**

\[
(x - 4)^2 + (y + 7)^2 = 8
\]

6. **Identify the center:**

The center of the circle is \((h, k) = (4, -7)\).

#### Correct Answer for Question 1:

- **a. (4, -7)**

### Question 2: Converting to Standard Form

The equation of the circle is:

\[
x^2 + y^2 + 10x - 4y - 7 = 0
\]

We will also rewrite this in the standard form \((x - h)^2 + (y - k)^2 = r^2\) by completing the square.

#### Step-by-Step Solution:

1. **Group the \(x\) and \(y\) terms:**

\[
(x^2 + 10x) + (y^2 - 4y) = 7
\]

2. **Complete the square for \(x\):**

\[
x^2 + 10x \rightarrow (x + 5)^2 - 25
\]

3. **Complete the square for \(y\):**

\[
y^2 - 4y \rightarrow (y - 2)^2 - 4
\]

4. **Substitute back into the equation:**

\[
(x + 5)^2 - 25 + (y - 2)^2 - 4 = 7
\]

5. **Combine constants:**

\[
(x + 5)^2 + (y - 2)^2 = 36
\]

6. **Identify the correct form:**

The standard form is \((x + 5)^2 + (y - 2)^2 = 6^2\).

#### Correct Answer for Question 2:

- **a. (x + 5)^2 + (y - 2)^2 = 62**

### Summary of Answers:

1. **Question 1:** a. (4, -7)
2. **Question 2:** a. (x + 5)^2 + (y - 2)^2 = 62

Would you like a more detailed explanation, or do you have any questions?

### Related Questions to Expand Your Understanding:

1. How do you complete the square for an equation with different coefficients?
2. What is the geometric significance of completing the square in a circle equation?
3. Can the center of a circle ever have negative coordinates?
4. How do changes in the equation of a circle affect its radius and position?
5. How do you find the radius of a circle from its standard form equation?

#### Tip:
Completing the square is a powerful technique to rewrite quadratic equations in a more useful form, such as finding the center and radius of a circle. Practice with different equations to become more comfortable with this process.

Which answer choice shows the center of the circle with the equation x^2 + y^2 - 8x + 14y + 57 = 0? What is the standard form of the equation of the circle x^2 + y^2 + 10x - 4y - 7 = 0?

Learn how to find the center of a circle from the equation x^2 + y^2 - 8x + 14y + 57 = 0, and how to convert x^2 + y^2 + 10x - 4y - 7 = 0 to standard form. This problem involves completing the square and understanding circle equations. Suitable for Grades 9-11 students.

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