General_Equation_of_a_Circle_Presentation_Updated.pptx

This updated presentation on the General Equation of a Circle provides an in-depth exploration of one of the fundamental curves in Analytic Geometry. The lesson focuses on understanding the relationship between the standard form and the general form of a circle’s equation, and how these forms can be interconverted through algebraic manipulation.

Students will learn how to analyze, derive, and interpret the equation of a circle given in different forms. Emphasis is placed on identifying the center and radius from the equation, completing the square to transform the general equation into standard form, and interpreting the geometric meaning of each component.

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 - 3 x - 3x + 9 + y² + 2 y + 2y + 4 = 25 x² - 6x + 9 + y² + 4y + 4 = 25 x² + y² - 6x + 4y + 9 + 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 : (x² - 4x) + (y² + 6y) = 12 C omplete the square: (x² - 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9    

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

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

12. 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    

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

14. 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    

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

16. 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

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

18. SW: Find the standard equation of the circle and sketch its graph. 1. Diameter with endpoints A (-3, 5) and B(5, -1) 2. Diameter with endpoints A (-4, -2) and B(2, 6)