This document outlines the objectives and formulas related to the equations of circles, discussing both the center-radius form and standard form of these equations. It provides multiple examples of how to derive the equations of circles based on given centers and radii, and how to identify the centers and radii from given equations. Additionally, it includes methods for converting equations from standard to general form.

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....
(x1, y1) = (1, 1)
(x2, y2) = (5, 2)
𝒅 = 𝒙𝟐 − 𝒙𝟏
𝟐 + (𝒚𝟐−𝒚𝟏)𝟐
𝒅 = 𝟓 − 𝟏 𝟐 + (𝟐 − 𝟏)𝟐
𝒅 = 𝟒𝟐 + 𝟏𝟐
𝒅 = 𝟏𝟕 𝒖𝒏𝒊𝒕𝒔
𝑫𝑰𝑺𝑻𝑨𝑵𝑪𝑬
FORMULA

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

9. center (0,0)
P(x,y)
𝒅 = 𝒙𝟐 − 𝒙𝟏
𝟐 + (𝒚𝟐−𝒚𝟏)𝟐
AP = r
(0,0)

10. 𝒙𝟐 − 𝒙𝟏
𝟐 + 𝒚𝟐 − 𝒚𝟏
𝟐 = 𝑨𝑷
𝒙 − 𝟎 𝟐 + 𝒚 − 𝟎 𝟐 = 𝒓
𝒙𝟐 + 𝒚𝟐 = 𝒓
𝒙𝟐
+ 𝒚𝟐
= 𝒓𝟐
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𝒙𝟐 + 𝒚𝟐
𝟐
= 𝒓 𝟐
A(x1, y1)= (0, 0) P(x2, y2) = (x, y)

11. Equation of a Circle in
Center - Radius Form
𝒙𝟐
+ 𝒚𝟐
= 𝒓𝟐

12. Example :
What is the equation of the
circle with center at the origin
and radius of length 8?
𝒙𝟐
+ 𝒚𝟐
= 𝒓𝟐
center = (0,0)
radius = 8 units 𝒙𝟐
+ 𝒚𝟐
= 𝟖𝟐
𝒙𝟐
+ 𝒚𝟐
= 𝟔𝟒
𝒙𝟐 + 𝒚𝟐 = 𝟔𝟒

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

14. center (h,k)
(x,y)
𝒅 = 𝒙𝟐 − 𝒙𝟏
𝟐 + 𝒚𝟐 −𝒚𝟏
𝟐
radius r

15. 𝒙𝟐 − 𝒙𝟏
𝟐 + 𝒚𝟐 − 𝒚𝟏
𝟐 = 𝒓
𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 = 𝒓
(𝒙 − 𝒉)𝟐
+(𝒚 − 𝒌)𝟐
= 𝒓𝟐
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(x1, y1) = (x2, y2) =
(h, k) (x, y)

16. Equation of a Circle in
Standard Form with
Center (h,k) and radius r
(𝒙 − 𝒉)𝟐
+ (𝒚 − 𝒌)𝟐
= 𝒓𝟐

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 = r2
[x – (-1)]2 + [y – (-1)]2 = 42
(x + 1)2 + (y + 1)2 = 42
(x + 1)2 + (y + 1)2 = 16
Given: C(h, k) = (–1, –1)

19. Equation of a Circle
in General Form

20. (𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐
(𝒙 − 𝒉)𝟐
= 𝒙𝟐
− 𝟐𝒉𝒙 + 𝒉𝟐
(𝒚 − 𝒌)𝟐
= 𝒚𝟐 − 𝟐𝒌𝒚 + 𝒌𝟐

21. (𝒙 − 𝒉)𝟐
+ (𝒚 − 𝒌)𝟐
= 𝒓𝟐
𝒙𝟐
− 𝟐𝒉𝒙 + 𝒉𝟐
+ 𝒚𝟐
− 𝟐𝒌𝒚 + 𝒌𝟐
= 𝒓𝟐
𝑳𝒆𝒕 𝑫 = −𝟐𝒉 𝑳𝒆𝒕 𝑬 = −𝟐𝒌
𝑳𝒆𝒕 𝑭 = 𝒉𝟐
+ 𝒌𝟐
− 𝒓𝟐
𝒙𝟐
− 𝟐𝒉𝒙 + 𝒉𝟐
+ 𝒚𝟐
− 𝟐𝒌𝒚 + 𝒌𝟐
−𝒓𝟐
= 𝟎
𝒙𝟐
+ 𝒚𝟐
+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

22. Equation of a Circle
in General Form
𝒙𝟐
+ 𝒚𝟐
+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

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.

25. (𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐
(𝒙 + 𝟐)𝟐 = 𝒙𝟐 + 𝟒𝒙 + 𝟒
(𝒚 − 𝟏)𝟐
= 𝒚𝟐
− 𝟐𝒚 + 𝟏
𝒄𝒆𝒏𝒕𝒆𝒓: ( −𝟐, 𝟏 ) 𝒓𝒂𝒅𝒊𝒖𝒔: 𝟔
[𝒙 − (−𝟐)]𝟐 + (𝒚 − 𝟏)𝟐 = 𝟔𝟐
(𝒙 + 𝟐)𝟐
+ (𝒚 − 𝟏)𝟐
= 𝟑𝟔

26. (𝒙 + 𝟐)𝟐 + (𝒚 − 𝟏)𝟐 = 𝟑𝟔
𝒙𝟐
+ 𝟒𝒙 + 𝟒 + 𝒚𝟐
− 𝟐𝒚 + 𝟏 = 𝟑𝟔
𝒙𝟐
+ 𝟒𝒙 + 𝟒 + 𝒚𝟐
− 𝟐𝒚 + 𝟏 − 𝟑𝟔 = 𝟎
𝒙𝟐
+ 𝒚𝟐
+ 𝟒𝒙 − 𝟐𝒚 − 𝟑𝟏 = 𝟎
𝒙𝟐
+ 𝒚𝟐
+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎

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
x2 + y2 = 24.

29. Given: x2 + y2 = 24
x2 + y2 = r2
center : (0,0)
radius = 2 𝟔 units
r2 = 24
𝒓𝟐 = 𝟐𝟒
r = 𝟐 𝟔
𝒓𝟐 = 𝟒(𝟔)
x2 + y2 = 24
(0,0)

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 = r2
(x – 3 )2 +
center (3, –2) radius = 7 units.
[ y -(-2 )]2
= 72
(y + 2)2 49

32. Example 3 :
Determine the center and
the radius of the circle
given by the equation
(x + 1)2 + y2 = 64.

33. Given: (x + 1)2 + y2 = 64
( x – h )2 + ( y - k )2 = r2
[x - (-1 )2 + (y - 0 )2 = 82
center : (-1,0) radius = 8 units

34. Example 4 :
Determine the center and
the radius of the circle
given by the equation
x2 + y2 – 6x – 135 = 0.

35. x2 + y2 – 6x – 135 = 0
x2 + y2 – 6x – 135 + 135 = 0 + 135
x2 + y2 – 6x = 135
(x2 – 6x ) + y2 = 135
(x2 – 6x + 9 ) + y2 = 135 + 9
(x – 3)2 + y2 = 144

36. ( x – h )2 + ( y – k )2 = r2
( x – 3 )2 + ( y – 0 )2 = 122
center : (3,0)
(x – 3)2 + y2 = 144
radius =12 units

37. Example 5:
Determine the
equation of the
circle whose
diameter has its
endpoints at
P1 (6, 2) and
P2 (4, 10).

38. 1. Find the midpoint of the diameter
to find the center of the circle.
P1 (6, 2) and P2 (4, 10)
(𝑥𝑚, 𝑦𝑚) =
𝑥1 + 𝑥2
2
,
𝑦1 + 𝑦2
2
=
6+4
2
,
2+10
2
= 5, 6
(𝑥1, 𝑦1) (𝑥2, 𝑦2)

39. 2. Determine the length of the radius r.
C(5, 6) and P2 (4, 10)
𝒓 = 𝒙𝟐 − 𝒙𝟏
𝟐 + 𝒚𝟐 − 𝒚𝟏
𝟐
𝒓 = 𝟒 − 𝟓 𝟐 + (𝟏𝟎 − 𝟔)𝟐
𝒓 = −𝟏 𝟐 + 𝟒𝟐
𝒓 = 𝟏 + 𝟏𝟔
𝒓 = 𝟏𝟕
(𝑥1, 𝑦1) (𝑥2, 𝑦2)
r
r

40. (x – h)2 + (y – k)2 = r2
(x – 5)2 + (y – 6)2 = ( 𝟏𝟕)2
(x – 5)2 + (y – 6)2 = 17
3. Substitute the values of C(h,k)=(5,6)
and the value of r = 𝟏𝟕
(x-5)2 + (y-5)2=17

41. Lyn On Me
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