General_Equation_of_a_Circle_Presentation_Updated (2).pptx

“General_Equation_of_a_Circle_Presentation_Updated” explains the concept and derivation of the general equation of a circle. It includes discussions on identifying the center and radius from the equation, converting between standard and general forms, and solving example problems for better understanding.

1. The General Equation of a Circle Understanding the Standard and General Forms Presented by: M s . R e v

2. Review – What is a Circle? A circle is the set of all points in a plane that are equidistant from a fixed point. - Center: the fixed point - Radius: the constant distance

3. Standard Form of a Circle (x - h)² + (y - k)² = r² Where: (h, k) = center of the circle r = radius Example: (x - 3)² + (y + 2)² = 25 Center: (3, -2) Radius: 5

4. General Equation of a Circle Where: - A , C , D , and E are constants - Comes from expanding the standard form

5. Converting Standard Equation into General Equation

6. Expanding the Standard Form Example: (x - 3)² + (y + 2)² = 25 (x - 3) ( x - 3 ) + (y + 2) ( y + 2 ) = 25 x² - 6x + 9 + y² + 4y + 4 = 25 x² + y² - 6x + 4y - 12 = 0

7. Converting General Equation into Standard Equation

8. Step-by-step Guide in Converting General Equation into Standard Equation Step 1: Group the terms. Step 2: Complete the Square Formula: Step 3: Find the binomial square.  

9. Identifying the Center and Radius Given: x² + y² - 4x + 6y - 12 = 0 Group terms and complete the square: (x² - 4x) + (y² + 6y) = 12 (x² - 4x) + 4 + (y² + 6y) + 9 = 12 + 4 + 9 (x - 2)² + (y + 3)² = 25 Center: (2, -3) Radius: 5

10. Practice Example 1 Convert to standard form and find center and radius: x² + y² + 8x - 10y + 31 = 0

11. Solution to Example 1 Group terms: (x² + 8x) + (y² - 10y) = -31 Complete the square: (x² + 8x + 16 ) + (y² - 10y + 25 ) = -31 + 16 + 25 Find the binomial square: (x + 4)² + (y - 5)² = 10 Center: (-4, 5) Radius: √10

12. Practice Example 2 Convert to standard form and find center and radius: x² + y² - 6x - 8y + 9 = 0

13. Solution to Example 2 Group terms: (x² - 6x) + (y² - 8y) = -9 Complete the square: (x² - 6x + 9 ) + (y² - 8y + 16 ) = -9 + 9 + 16 Find the binomial square: (x - 3)² + (y - 4)² = 16 Center: (3, 4) Radius: 4

14. Practice Example 3 Convert to standard form and find center and radius: x² + y² + 2x - 12y + 20 = 0

15. Solution to Example 3 Group terms: (x² + 2x) + (y² - 12y) = -20 Complete the square: (x² + 2x + 1 ) + (y² - 12y + 36 ) = -20 + 1 + 36 Find the binomial square: (x + 1)² + (y - 6)² = 17 Center: (-1, 6) Radius: √17

16. Try It Yourself! Convert to standard form and find the center and radius: x² + y² - 2x + 4y - 4 = 0