10.8 Day 2 Extension

Example 1: Give the center and radius of the circle. a) (𝑥−3 ) 2 +(𝑦+11 ) 2 =49 b) (𝑥−4 ) 2 +(𝑦−1 ) 2 =25   Center: ___________ Center: ____________ Radius: ___________ Radius: ____________ (3, −11) (4, 1) 7 5

Example 2: Give the center, radius, and equation of the circle. Center: _______ b) Center: _______ Radius: _______ Radius: _______ (5, −2) (1, 1) 3 2

Example 3: Write the standard equation of the circle with the given center and radius/diameter. Center: (-1.5, 0) b)Center: (6, -2) Radius: 2 Diameter: 8 Equation: Equation: (𝑥+1.5 ) 2 + 𝑦 2 =4 (𝑥−36+(𝑦+2 ) 2 =64

Example 4: Write the equation of a circle that has its center at (-1, -3) and passes through (2, 1). (2+1) 2 + (1+3) 2 = 9+16 = 25 =5 (𝑥+1 ) 2 +(𝑦+3 ) 2 =25

Example 5: Write the equation of a circle that has its center at the origin and passes through (9, 12). (9−0) 2 + (12−0) 2 = 81+144 = 225 =15 𝑥 2 + 𝑦 2 =225

Example 6: Write the equation of a circle that has its center at (-3, 3) and passes through (-6, 2). (−6+3) 2 + (2−3) 2 = 9+1 = 10 (𝑥+3 ) 2 +(𝑦−3 ) 2 =10

Completing the square The following steps were used to convert from general to standard.  Arrange the x terms together and the y terms together and move the constant to the other side. Complete the square for the x's and for the y's. Balance the equation by adding the numbers found to the other side as well. Write each binomial squared and combine the numbers. Identify the radius and center of the circle.

Write each equation of a circle in standard form Write each equation of a circle in standard form. Then identify the center and radius. Example 7: 𝑥 2 + 𝑦 2 −10𝑥−20𝑦−2=0 𝑥 2 −10𝑥+ 𝑦 2 −20𝑦=2 ( −10 2 ) 2 =25 ( −20 2 ) 2 =100 𝑥 2 −10𝑥+25+ 𝑦 2 −20𝑦+100=2+25+100 (𝑥−5) 2 + (𝑦−10) 2 =127 Center = (5, 10) Radius = 127

Write each equation of a circle in standard form Write each equation of a circle in standard form. Then identify the center and radius. Example 8: 𝑥 2 + 𝑦 2 −4𝑥+6𝑦−12=0 𝑥 2 −4𝑥+ 𝑦 2 +6𝑦=12 ( −4 2 ) 2 =4 ( 6 2 ) 2 =9 𝑥 2 −4𝑥+4+ 𝑦 2 +6𝑦+9=12+4+9 (𝑥−2) 2 + (𝑦+3) 2 =25 Center = (2, -3) Radius = 5

Write each equation of a circle in standard form Write each equation of a circle in standard form. Then identify the center and radius. Example 9: 𝑥 2 + 𝑦 2 +2𝑥−4𝑦−4=0 𝑥 2 +2𝑥+ 𝑦 2 −4𝑦=4 ( 2 2 ) 2 =1 ( −4 2 ) 2 =4 𝑥 2 +2𝑥+1+ 𝑦 2 −4𝑦+4=4+1+4 (𝑥+1) 2 + (𝑦−2) 2 =9 Center = (-1, 2) Radius = 3

Example 10: The equation of a circle is x2 – 4x + y2 + 6y = –9. State the coordinates of the center and the measure of the radius. Then graph the equation. 𝑥 2 −4𝑥+ 𝑦 2 +6𝑦=−9 ( −4 2 ) 2 =4 ( 6 2 ) 2 =9 𝑥 2 −4𝑥+4+ 𝑦 2 +6𝑦+9=−9+4+9 (𝑥−2) 2 + (𝑦+3) 2 =4 Center = (2, -3) Radius = 2

Example 11: Jimmy Johns offers delivery within 5 miles of the restaurant. Marcus’ house is located 4 miles east and 5 miles south of Jimmy Johns. Carly’s house is located 1 mile west and 2 miles south of Jimmy Johns. Conner’s house is located 3 miles east and 5 miles north of Jimmy Johns. If Jimmy Johns is located at (–2, 1), which house(s) can get delivery? Since Carly’s house is the only house inside of the 5 mile radius around Jimmy Johns, her house is the only house than can get delivery. Conner Carly Marcus

Example 12: Cell phone towers are periodically placed around App Town so that they give a signal that reaches within 4 miles of the tower’s location. One particular tower is located at (3, 4). Houses A, B, and C are located at (–1, 4), (5, 3), and (2, 0). Which house(s) receive a signal from this tower? Houses A and B are within the tower’s location so they will receive a signal from this tower. A B C