https://file.aiquickdraw.com/mathbot/file/1726858180396oq8uu3u3.jpg

The problem asks to convert the given center-radius form of a circle's equation into the general form. The provided equation is:

\[
(x+5)^2 + (y-3)^2 = 12
\]

To convert this to the general form \(x^2 + y^2 + ax + by + c = 0\), we need to expand and simplify the given equation:

1. Expand \( (x+5)^2 \) and \( (y-3)^2 \):
   \[
   (x+5)^2 = x^2 + 10x + 25
   \]
   \[
   (y-3)^2 = y^2 - 6y + 9
   \]

2. Add these expansions and set equal to 12:
   \[
   x^2 + 10x + 25 + y^2 - 6y + 9 = 12
   \]

3. Combine and simplify constants:
   \[
   x^2 + 10x + y^2 - 6y + 25 + 9 - 12 = 0
   \]
   \[
   x^2 + 10x + y^2 - 6y + 22 = 0
   \]

Thus, the equation in general form is:
\[
x^2 + y^2 + 10x - 6y + 22 = 0
\]

Among the given options:

- A. \(x^2 + y^2 + 5x - 3y - 12 = 0\)
- B. \(x^2 + y^2 + 5x - 3y + 22 = 0\)
- C. \(x^2 + y^2 + 10x - 6y + 22 = 0\)
- D. \(x^2 + y^2 + 10x - 6y - 12 = 0\)

Option C is the correct one.

Would you like more details or help with another question? Here are some other things you might want to explore:
1. How to derive the center and radius from the general form of a circle equation.
2. What changes if the center of the circle is shifted?
3. How to graph the equation of a circle given in general form.
4. How to find the intersection of this circle with a line.
5. How transformations like translations affect the equation of a circle.

**Tip**: When converting a circle's equation from center-radius form to general form, it's useful to fully expand and then simplify the terms step-by-step to avoid errors.

The equation of a circle is given in center-radius form. Write the equation in general form: (x + 5)^2 + (y - 3)^2 = 12.

Learn how to convert the circle equation (x + 5)^2 + (y - 3)^2 = 12 from center-radius form to general form, step by step. Understand key geometric and algebraic concepts suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation