Let's focus on question 15, which involves converting the equation of a circle from its General Form to its Standard Form.
### Step-by-Step Solution:
1. Equation in General Form:
The given equation of the circle in General Form is:
[tex]\[
x^2 + y^2 - 2x + 6y - 6 = 0
\][/tex]
2. Grouping the Terms:
Group the [tex]\(x\)[/tex] terms and [tex]\(y\)[/tex] terms together:
[tex]\[
(x^2 - 2x) + (y^2 + 6y) = 6
\][/tex]
3. Completing the Square on the [tex]\(x\)[/tex] Terms:
- Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-2\)[/tex], halve it to get [tex]\(-1\)[/tex], and then square it to get [tex]\(1\)[/tex].
- Add and subtract [tex]\(1\)[/tex] to complete the square:
[tex]\[
(x^2 - 2x + 1 - 1) = (x - 1)^2 - 1
\][/tex]
4. Completing the Square on the [tex]\(y\)[/tex] Terms:
- Take the coefficient of [tex]\(y\)[/tex], which is [tex]\(6\)[/tex], halve it to get 3, and then square it to get 9.
- Add and subtract 9 to complete the square:
[tex]\[
(y^2 + 6y + 9 - 9) = (y + 3)^2 - 9
\][/tex]
5. Rewriting the Equation with Completed Squares:
Substitute the completed square terms back into the equation:
[tex]\[
(x - 1)^2 - 1 + (y + 3)^2 - 9 = 6
\][/tex]
6. Simplifying the Equation:
Combine constants on the right side to convert to the Standard Form:
[tex]\[
(x - 1)^2 + (y + 3)^2 - 10 = 6
\][/tex]
[tex]\[
(x - 1)^2 + (y + 3)^2 = 16
\][/tex]
7. Conclusion:
The equation in Standard Form is:
[tex]\[
(x - 1)^2 + (y + 3)^2 = 16
\][/tex]
### Final Answer:
Therefore, the equation of the circle in Standard Form is:
[tex]\[
B. (x - 1)^2 + (y + 3)^2 = 16
\][/tex]
So, the correct option is B.
