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

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.

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Center-Radius Form of the Equation of a Circle
- 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.
(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
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
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
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
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
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
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?
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.