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.