This document discusses the center-radius form of the equation of a circle. It provides the standard equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) are the coordinates of the center and r is the radius. It also shows how to determine the center and radius from a given equation, either in standard or general form. Examples are provided to illustrate writing the equation in both standard and general form, as well as determining the center and radius from equations in various forms.

1. Center-Radius
Form of the
Equation of a
Circle

2. - Illustrates the center-radius form of the
equation of a circle.
- Determine the center and radius of a
circle given its equation and vice versa.

5. (𝑥, 𝑦)
𝑟 = (𝑥 − ℎ)2+(𝑦 − 𝑘)2
𝑟2
= (𝑥 − ℎ)2+(𝑦 − 𝑘)2
2
𝑟2
= (𝑥 − ℎ)2
+(𝑦 − 𝑘)2
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
The equation of the circle with center (ℎ, 𝑘) and radius 𝑟 is
given by
Standard Form
𝑐
(ℎ, 𝑘)
𝑟

6. A point (𝑥, 𝑦) can be one of the following depending on
which equation or inequality the coordinates satisfy:
On the circle
Interior of
the circle
Exterior of
the circle
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
< 𝑟2
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
> 𝑟2

7. When the standard form (𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
is expanded,
the equation become
𝑥2
− 2ℎ𝑥 + ℎ2
+ 𝑦2
− 2𝑘𝑦 + 𝑘2
− 𝑟2
= 0
𝑥2
+ 𝑦2
+ −2ℎ 𝑥 + (−2𝑘)𝑦 + ℎ2
+ 𝑘2
− 𝑟2
= 0
𝐴𝑥2
+ 𝐵𝑦2
+ 𝐶𝑥 + 𝐷𝑦 + 𝐶 = 0
General Form
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐵 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑧𝑒𝑟𝑜

8. Example 1: Write equation on both standard and
general form.
a. C (0, 0) and 𝑟 =5
b.C (-1, 2) and 𝑟 = 2 5
Try This
Write the equation of the circle in both standard and
general forms.
a. C (0, 0) and 𝑟 = 3
b.C (-2, -3) and 𝑟 = 6

9. Example 2: Identify the coordinates of the center and
the length of the radius of the circle from the given
equation of the circle.
1.𝑥2
+ 𝑦2
= 9
2.(𝑥 + 1)2
+ 𝑦 − 2 2
= 12
3.𝑥2
+ 𝑦2
− 6𝑥 + 2𝑦 + 6 = 0

10. Try This
Identify the coordinates of the center and the length of
the radius of the circle from the given equation of the
circle.
1. 𝑥2
+𝑦2
− 1 = 0
2.(𝑥 − 5)2
+ 𝑦 − 4 2
= 10
3. 𝑥2
+𝑦2
− 2𝑥 + 𝑦 − 1 = 0

11. Reflect:
“Gulong ng Palad” (literally translated as Wheel of the Palm or
figuratively Wheel of Fate) is a famous Filipino saying or
proverbs goes “Ang buhay ay parang gulong, minsan nasa
ibabaw, minsan nasa ilalim.”. What does this mean? Give
example. How then should we live our life?

12. Enrichment Activity
A. TRUE or FALSE #1-15
B. Describe whether the point is in the interior, exterior, on
the circle or none of these.
Kindly check ToHS Math 10 for your activity.