1. Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )

2. Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r ) The distance from the center of the circle to any point ( x, y ) ON the circle is the RADIUS

3. Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r ) When the equation of the circle is given in the form; You must rewrite the equation in standard form by completing the square…

4. Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r ) When the equation of the circle is given in the form; You must rewrite the equation in standard form by completing the square… Let’s look at the standard form first…

5. Distance and Circles ( h, k ) r Find the center and radius of the circle whose equations is :

6. Distance and Circles ( h, k ) r Find the center and radius of the circle whose equations is : To get ( x – 4 ), h would have to be +4 - ( x – h ) 2 = ( x – (+4)) 2 = (x – 4 ) 2

7. Distance and Circles ( h, k ) r Find the center and radius of the circle whose equations is : To get ( x – 4 ), h would have to be +4 - ( x – h ) 2 = ( x – (+4)) 2 = (x – 4 ) 2 To get ( y + 3 ), k would have to be - 3 - ( y – k ) 2 = ( y – ( -3)) 2 = ( y + 3 ) 2

8. Distance and Circles ( h, k ) r Find the center and radius of the circle whose equations is : To get ( x – 4 ), h would have to be +4 - ( x – h ) 2 = ( x – (+4)) 2 = (x – 4 ) 2 To get ( y + 3 ), k would have to be - 3 - ( y – k ) 2 = ( y – ( -3)) 2 = ( y + 3 ) 2 CENTER = ( 4, - 3 )

9. Distance and Circles ( h, k ) r Find the center and radius of the circle whose equations is : To get ( x – 4 ), h would have to be +4 - ( x – h ) 2 = ( x – (+4)) 2 = (x – 4 ) 2 To get ( y + 3 ), k would have to be - 3 - ( y – k ) 2 = ( y – ( -3)) 2 = ( y + 3 ) 2 There is a short cut…just use the OPPOSITE sign you see in front of h and k CENTER = ( 4, - 3 )

10. Distance and Circles ( h, k ) r Find the center and radius of the circle whose equations is : To get ( x – 4 ), h would have to be +4 - ( x – h ) 2 = ( x – (+4)) 2 = (x – 4 ) 2 To get ( y + 3 ), k would have to be - 3 - ( y – k ) 2 = ( y – ( -3)) 2 = ( y + 3 ) 2 There is a short cut…just use the OPPOSITE sign you see in front of h and k CENTER = ( 4, - 3 ) and if r 2 = 36, r = 6

11. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial

12. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation :

13. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Rewrite the equation getting your x’s and y’s together.

14. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Rewrite the equation getting your x’s and y’s together. Move any integer to the other side of the equation.

15. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Rewrite the equation getting your x’s and y’s together. Move any integer to the other side of the equation. Leave one blank space behind each x/y group and 2 behind your #

16. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Write the standard equation form leaving blanks in the spots in squares… also leave a few lines space for the next step in between…

17. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : To complete the square, divide the linear x and y coefficient by 2…

18. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : To complete the square, divide the linear x and y coefficient by 2…the answer will fill in the blank spaces in the standard form…

19. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Next, square those answers and fill in the blank spaces on both sides of the equation…

20. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Next, square those answers and fill in the blank spaces on both sides of the equation…

21. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Then, complete the addition on the right side and fill in the the last blank in the standard form…

22. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Then, complete the addition on the right side and fill in the the last blank in the standard form…

23. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Let’s clean up our double signs….

24. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Center = ( - 8, - 5 ) r = 3

25. Distance and Circles Completing the square – forcing an expression into a perfect square trinomial EXAMPLE : Find the center and radius of a circle defined by the equation : Center = ( - 8, - 5 ) r = 3

26. EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

32. Now complete your square…

33. EXAMPLE #2 : Find the center and radius of a circle defined by the equation : Now complete your square…

34. EXAMPLE #2 : Find the center and radius of a circle defined by the equation : Fill in the squares…

35. EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

40. EXAMPLE #3 : Find the center and radius of a circle defined by the equation :

44. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) :

45. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) : 1. Begin by substituting ( h, k ) into our circle equation :

46. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) :

47. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) :

48. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) :

49. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) :

50. EXAMPLE #2 : Find the equation of the circle whose center is ( 4, 5 ) and goes thru the coordinate ( - 1, 6 ) :