Circle Equation PDF

Kerr MPM2D Unit 2: Line Segments and Circles 1 2.3 Equation of a Circle Teacher Copy A circle is t...

Kerr MPM2D Unit 2: Line Segments and Circles 1 2.3 Equation of a Circle Teacher Copy A circle is the set of all points on a plane that are the same distance from a fixed point, the center. When the center of the circle is on the origin, the equation of the circle is 𝑥 2 + 𝑦 2 = 𝑟 2. When the center of the circle is NOT on the origin, the equation of the circle is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2. Examples 1a) Write the equation of the circle in the diagram. i) ii) 𝑥2 + 𝑦2 = 𝑟2 (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 𝑥 2 + 𝑦 2 = 42 (𝑥 + 4)2 + (𝑦 + 3)2 = 52 𝑥 2 + 𝑦 2 = 16 (𝑥 + 4)2 + (𝑦 + 3)2 = 25 b) State the x-intercepts. (4, 0) and (-4, 0) c) State the y-intercepts. (0, 4) and (0, -4) d) State the radius. r=4  Kerr MPM2D Unit 2: Line Segments and Circles 2 2. What is the radius for the circle defined by 𝑥 2 + 𝑦 2 = 289? 𝑟 2 = 289 𝑟 = 17 3a) Write the equation of a circle with center (0, 0) and a radius of 11. 𝑥 2 + 𝑦 2 = 112 𝑥 2 + 𝑦 2 = 121 b) Besides the x-intercepts and y-intercepts, state two other ordered pairs on the circle. Express your answer in radical form. First isolate y. 𝑦2 = 𝑟2 − 𝑥2 𝑦 = √𝑟 2 − 𝑥 2 The radius is 11. Now choose any value of x that is less than the length of the radius. I chose 8. 𝑦 = √112 − 82 𝑦 = √121 − 64 𝑦 = √57 Therefore, the points (8, √57) 𝑎𝑛𝑑 (8, −√57) are on the circle. 4a) Write the equation of a circle with center (0, 0) and a radius of 14. 3 14 𝑥 2 + 𝑦 2 = ( )2 3 196 𝑥2 + 𝑦2 = 9 14 14 b) State the x-intercepts. ( 3 , 0) 𝑎𝑛𝑑 (− , 0) 3 14 14 c) State the y-intercepts. (0, 3 ) 𝑎𝑛𝑑 (0, − 3 ) d) Sketch a diagram.  Kerr MPM2D Unit 2: Line Segments and Circles 3 5. A circle has a center of (0, 0) and passes through the point (5, -7). a) Calculate the radius and sketch the graph. Round to the nearest hundredth. 𝑟2 = 𝑥2 + 𝑦2 𝑟 2 = (5)2 + (−7)2 𝑟 2 = 25 + 49 𝑟 2 = 74 𝑟 = √74 𝑟 = 8.60 b) Does the point (-8, -2) lie inside, outside or on the circle? 𝑟 2 = (−8)2 + (−2)2 𝑟 2 = 64 + 4 𝑟 2 = 68 Since 68 is less than 74, the point lies inside the circle. c) Write the equation of the circle. 𝑥 2 + 𝑦 2 = 74  Kerr MPM2D Unit 2: Line Segments and Circles 4 6. A stone is dropped into a pond, creating a circular ripple. The radius of the ripple increases steadily at 15.9 cm/second. A lily pad is floating on the pond, 3.5 m east and 2.25 m south of the spot where the stone is dropped. How long does the ripple take to reach the lily pad? Round to the nearest hundredth. 3.5 m = 350 cm 2.25 m = 225 cm 𝑟2 = 𝑥2 + 𝑦2 𝑟 2 = 3502 + 2252 𝑟 2 = 173125 𝑟 = 416.08 𝑐𝑚 416.08 ÷ 15.9 = 26.17 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 Therefore, it takes 26.17 seconds for the ripple to reach the lily pad. Homework Page 91 #1, 2, 3c, 4-8, 10-15, 18a Special Instructions For question #10 you must calculate circumference (𝐶 = 2𝜋𝑟) Question #13 is very similar to example #6.