Equation of a Circle in standard and general form

Writing an equation of a circle in standard or general form and finding the radius and the center of a circle given an equation.

1. WELCOME 10-Burgos 10-Dagohoy 10-Del Pilar

2. QUARTER 2 WEEK 8

3. OBJECTIVES

4. Illustrate the center-radius form of the equation of a circle.

5. Determine the center and radius of a circle given its equation and vice versa.

6. Look Back to your Lesson.... Look Back to your Lesson....

7. Look Back to your Lesson.... ( x 1 , y 1 ) = (1, 1) ( x 2 , y 2 ) = (5, 2)         FORMULA  

8. Equation of a Circle in Standard Form or Center-Radius Form

9. center (0,0) P (x,y)   AP = r (0,0)

10.         Lyn On Me   A( x 1 , y 1 )= (0, 0) P(x 2 , y 2 ) = (x, y)

11. Equation of a Circle in Center - Radius Form  

12. Example : What is the equation of the circle with center at the origin and radius of length 8?   center = (0,0) radius = 8 units      

13. Equation of a Circle with Center ( h,k ) and Radius r

14. center (h,k) (x,y)   radius r

15.       Lyn On Me ( x 1 , y 1 ) = ( x 2 , y 2 ) = (h, k) (x, y)

16. Equation of a Circle in Standard Form with Center ( h,k ) and radius r  

17. Find the equation of a circle, in standard form, whose center is (–1, –1), and radius of 4 units. Example ( -1,-1 ) 4

18. r = 4 units (x – h) 2 + (y – k) 2 = r 2 [x – ( -1 )] 2 + [y – ( -1 )] 2 = 4 2 (x + 1) 2 + (y + 1) 2 = 4 2 (x + 1) 2 + (y + 1) 2 = 16 Given: C(h, k) = (–1, –1)

19. Equation of a Circle in General Form

21.   =           +  

22. Equation of a Circle in General Form +  

23. Equation of a Circle in General Form a. Standard form b. Square the binomial d. Simplify c. Transpose

24. Example Write the equation of a circle, in general form, with coordinates of the center at (-2, 1) and radius 6.

26.         +  

27. Determine the center and radius of the circle given an equation .

28. Example 1: Determine the center and radius of the circle given by the equation x 2 + y 2 = 24.

29. Given: x 2 + y 2 = 24 x 2 + y 2 = r 2 center : (0,0) radius = 2 units   r 2 = 24 =   r =   =   x 2 + y 2 = 24 (0,0) 2  

30. Example 2 : Determine the center and radius of the circle given by the equation (x - 3) 2 + (y + 2) 2 = 49.

31. Given: (x – 3) 2 + (y + 2) 2 = 49 ? (x – h ) 2 + (y – k ) 2 = r 2 (x – 3 ) 2 + center (3, –2) radius = 7 units. [ y -(-2 )] 2 = 7 2 (y + 2) 2 49

32. Example 3 : Determine the center and the radius of the circle given by the equation (x + 1) 2 + y 2 = 64.

33. Given: (x + 1) 2 + y 2 = 64 ( x – h ) 2 + ( y - k ) 2 = r 2 [ x - ( -1 ) 2 + (y - ) 2 = 8 2 center : (-1,0) radius = 8 units

34. Example 4 : Determine the center and the radius of the circle given by the equation x 2 + y 2 – 6x – 135 = 0.

35. x 2 + y 2 – 6x – 135 = 0 x 2 + y 2 – 6x – 135 + 135 = 0 + 135 x 2 + y 2 – 6x = 135 (x 2 – 6x ) + y 2 = 135 (x 2 – 6x + 9 ) + y 2 = 135 + 9 (x – 3) 2 + y 2 = 144

36. ( x – h ) 2 + ( y – k ) 2 = r 2 ( x – 3 ) 2 + ( y – 0 ) 2 = 12 2 center : (3,0) (x – 3) 2 + y 2 = 144 radius = 12 units

37. Example 5: Determine the equation of the circle whose diameter has its endpoints at P 1 (6, 2) and P 2 (4, 10).

38. 1. Find the midpoint of the diameter to find the center of the circle. P 1 ( 6, 2 ) and P 2 ( 4, 10 )          

39. 2. Determine the length of the radius r. C(5, 6) and P 2 (4, 10)               r r

40. (x – h) 2 + (y – k) 2 = r 2 ( x – 5 ) 2 + ( y – 6 ) 2 = ) 2 ( x – 5 ) 2 + ( y – 6 ) 2 = 17   3. Substitute the values of C(h,k)=(5,6) and the value of r =   (x-5) 2 + (y-5) 2 =17

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