10-7: Write and Graph Equations of Circles Geometry Chapter 10 10-7: Write and Graph Equations of Circles

Warm-Up Find the distance between the points given. 1.) (−4, 2) and (2, 1) 2.) (4, 4) and (−3, −3) 3.) (3, −2) and (−5, −2) 4.) (5, 3) and (5, 10)

Warm-Up Find the distance between the points given. 1.) (−𝟒, 𝟐) and (𝟐, 𝟏) 2.) (𝟒, 𝟒) and (−𝟑, −𝟑) 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑑= (2−(−4)) 2 + (1−2) 2 𝑑= (−3 −4) 2 + (−3−4) 2 𝑑= (6) 2 + (−1) 2 𝑑= (−7) 2 + (−7) 2 𝑑= 36+1 = 37 𝑑= 49+49 = 98 𝒅≈𝟔.𝟎𝟖𝟑 𝒅≈𝟗.𝟖𝟗𝟗

Warm-Up Find the distance between the points given. 3.) (𝟑, −𝟐) and (−𝟓, −𝟐) 4.) (𝟓, 𝟑) and (𝟓, 𝟏𝟎) 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑑= (−5−3) 2 + (−2−(−2)) 2 𝑑= (5−5) 2 + (10−3) 2 𝑑= (−8) 2 + (0) 2 𝑑= (0) 2 + (7) 2 𝑑= 64 𝑑= 49 𝒅=𝟖 𝒅=𝟕

Write and Graph Equations of Circles Objective: Students will be able to write equations of circles and graph them on the coordinate plane. Agenda Circle Equation Graphs of Circles

Circles – Review Recall that for a circle… Center: The point in the plane that all points of the circle are equidistant to. Radius: The line that represents the distance from any given point on the circle to the center. Center 𝒓

Circles – Review Recall that for a circle… Center: The point in the plane that all points of the circle are equidistant to. Radius: The line that represents the distance from any given point on the circle to the center. Since the radius can be expressed as the distance between the center and all points around it, we can use the distance formula to make an equation for the circle… Center 𝒓

Equation of a Circle An equation of a circle with center (𝑎, 𝑏) and radius r is (𝒙−𝒂) 𝟐 + (𝒚−𝒃) 𝟐 = 𝒓 𝟐

Equation of a Circle An equation of a circle with center (𝑎, 𝑏) and radius r is (𝒙−𝒂) 𝟐 + (𝒚−𝒃) 𝟐 = 𝒓 𝟐 Example 1: Write the equation of a circle with the given center and radius Center: 𝟐, 𝟓 ; 𝒓𝒂𝒅𝒊𝒖𝒔=𝟑

Equation of a Circle An equation of a circle with center (𝑎, 𝑏) and radius r is (𝒙−𝒂) 𝟐 + (𝒚−𝒃) 𝟐 = 𝒓 𝟐 Example 1: Write the equation of a circle with the given center and radius Center: 𝟐, 𝟓 ; 𝐫𝐚𝐝𝐢𝐮𝐬=𝟑 Solution: (𝒙−𝒂) 𝟐 + (𝒚−𝒃) 𝟐 = 𝒓 𝟐 (𝒙−𝟐) 𝟐 + (𝒚−𝟓) 𝟐 = 𝟑 𝟐 (𝒙−𝟐) 𝟐 + (𝒚−𝟓) 𝟐 =𝟗

Example 2 Write the equation of a circle with the given center and radius a.) Center: 0, −9 ; 𝑟𝑎𝑑𝑖𝑢𝑠=4.2 b.) Center: −2, 5 ; 𝑟𝑎𝑑𝑖𝑢𝑠=7 c.) Center: 0, 0 ; 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟=10

Example 2 Write the equation of a circle with the given center and radius a.) Center: 0, −9 ; 𝑟𝑎𝑑𝑖𝑢𝑠=4.2 b.) Center: −2, 5 ; 𝑟𝑎𝑑𝑖𝑢𝑠=7 (𝑥−0) 2 + (𝑦−(−9)) 2 = 4.2 2 𝒙 𝟐 + (𝒚+𝟗) 𝟐 =𝟏𝟕.𝟔𝟒 c.) Center: 0, 0 ; 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟=10

Example 2 Write the equation of a circle with the given center and radius a.) Center: 0, −9 ; 𝑟𝑎𝑑𝑖𝑢𝑠=4.2 b.) Center: −2, 5 ; 𝑟𝑎𝑑𝑖𝑢𝑠=7 (𝑥−0) 2 + (𝑦−(−9)) 2 = 4.2 2 (𝑥−(−2)) 2 + (𝑦−5) 2 = 7 2 𝒙 𝟐 + (𝒚+𝟗) 𝟐 =𝟏𝟕.𝟔𝟒 (𝒙+𝟐) 𝟐 + (𝒚−𝟓) 𝟐 =𝟒𝟗 c.) Center: 0, 0 ; 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟=10

Example 2 Write the equation of a circle with the given center and radius a.) Center: 0, −9 ; 𝑟𝑎𝑑𝑖𝑢𝑠=4.2 b.) Center: −2, 5 ; 𝑟𝑎𝑑𝑖𝑢𝑠=7 (𝑥−0) 2 + (𝑦−(−9)) 2 = 4.2 2 (𝑥−(−2)) 2 + (𝑦−5) 2 = 7 2 𝒙 𝟐 + (𝒚+𝟗) 𝟐 =𝟏𝟕.𝟔𝟒 (𝒙+𝟐) 𝟐 + (𝒚−𝟓) 𝟐 =𝟒𝟗 c.) Center: 0, 0 ; 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟=10 (𝑥−0) 2 + (𝑦−0) 2 = 5 2 𝒙 𝟐 + 𝒚 𝟐 = 𝟐𝟓 𝟐

Example 3 Write the standard equation of the circle shown on the graph. a.)

Example 3 Write the standard equation of the circle shown on the graph. a.) You can see that the center of the circle is at the origin (0, 0), and the radius is 3 units. Thus,

Example 3 Write the standard equation of the circle shown on the graph. a.) You can see that the center of the circle is at the origin (0, 0), and the radius is 3 units. Thus, (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−0) 2 + (𝑦−0) 2 = 3 2 𝒙 𝟐 + 𝒚 𝟐 =𝟗

Example 3 Write the standard equation of the circle shown on the graph. b.)

Example 3 Write the standard equation of the circle shown on the graph. b.) You can see that the center of the circle is at the origin (0, 0), and the radius is 4 units. Thus, (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−0) 2 + (𝑦−0) 2 = 4 2 𝒙 𝟐 + 𝒚 𝟐 =𝟏𝟔

Example 3 Write the standard equation of the circle shown on the graph. c.)

Example 3 Write the standard equation of the circle shown on the graph. c.) You can see that the center of the circle is at the origin (0, 0), and the radius is 2 units. Thus, (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−0) 2 + (𝑦−0) 2 = 2 2 𝒙 𝟐 + 𝒚 𝟐 =𝟒

Example 3 Write the standard equation of the circle shown on the graph. d.)

Example 3 Write the standard equation of the circle shown on the graph. d.) You can see that the center of the circle is at the point (2, 3), and the radius is 2 units. Thus, (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−2) 2 + (𝑦−3) 2 = 2 2 (𝒙−𝟐) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟒

Example 4 The point (−𝟓, 𝟔) is on a circle with center (−𝟏, 𝟑). Write the standard equation of the circle.

Example 4 The point (−𝟓, 𝟔) is on a circle with center (−𝟏, 𝟑). Write the standard equation of the circle. We can use the distance formula to find the radius: 𝑟= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑟= (−1−(−5)) 2 + (3−6) 2 𝑟= (4) 2 + (−3) 2 𝒓= 𝟏𝟔+𝟗 = 𝟐𝟓 =𝟓

Example 4 The point (−𝟓, 𝟔) is on a circle with center (−𝟏, 𝟑). Write the standard equation of the circle. Thus, the standard equation is (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−(−1)) 2 + (𝑦−3) 2 = 5 2 (𝒙+𝟏) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟐𝟓

Example 5 a.) The point (𝟑, 𝟒) is on a circle with center (𝟏, 𝟒). Write the standard equation of the circle.

Example 5 a.) The point (𝟑, 𝟒) is on a circle with center (𝟏, 𝟒). Write the standard equation of the circle. We can use the distance formula to find the radius: 𝑟= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑟= (1−3) 2 + (4−4) 2 𝑟= (−2) 2 + (0) 2 𝒓= 𝟒+𝟎 = 𝟒 =𝟐

Example 5 a.) The point (𝟑, 𝟒) is on a circle with center (𝟏, 𝟒). Write the standard equation of the circle. Thus, the standard equation is (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−1) 2 + (𝑦−4) 2 = 2 2 (𝒙−𝟏) 𝟐 +( 𝒚−𝟒) 𝟐 =𝟒

Example 5 b.) The point (−𝟏, 𝟐) is on a circle with center (𝟐, 𝟔). Write the standard equation of the circle.

Example 5 b.) The point (−𝟏, 𝟐) is on a circle with center (𝟐, 𝟔). Write the standard equation of the circle. We can use the distance formula to find the radius: 𝑟= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝑟= (2−(−1)) 2 + (6−2) 2 𝑟= (3) 2 + (4) 2 𝒓= 𝟗+𝟏𝟔 = 𝟐𝟓 =𝟓

Example 5 b.) The point (−𝟏, 𝟐) is on a circle with center (𝟐, 𝟔). Write the standard equation of the circle. Thus, the standard equation is (𝑥−𝑎) 2 + (𝑦−𝑏) 2 = 𝑟 2 (𝑥−2) 2 + (𝑦−6) 2 = 5 2 (𝒙−𝟐) 𝟐 +( 𝒚−𝟔) 𝟐 =𝟐𝟓

Example 6a The standard equation of a circle is (𝒙−𝟒) 𝟐 +( 𝒚+𝟐) 𝟐 =𝟑𝟔. Graph the circle in the space provided.

Example 6a The standard equation of a circle is (𝒙−𝟒) 𝟐 +( 𝒚+𝟐) 𝟐 =𝟑𝟔. Graph the circle in the space provided. Center: (𝟒, −𝟐)

Example 6a The standard equation of a circle is (𝒙−𝟒) 𝟐 +( 𝒚+𝟐) 𝟐 =𝟑𝟔. Graph the circle in the space provided. Center: (𝟒, −𝟐) Radius: 𝒓= 𝟑𝟔 =𝟔

Example 6b The standard equation of a circle is (𝒙+𝟏) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟒. Graph the circle in the space provided.

Example 6b The standard equation of a circle is (𝒙+𝟏) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟒. Graph the circle in the space provided. Center: (−𝟏, 𝟑)

Example 6b The standard equation of a circle is (𝒙+𝟏) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟒. Graph the circle in the space provided. Center: (−𝟏, 𝟑) Radius: 𝒓= 𝟒 =𝟐

Example 6c The standard equation of a circle is 𝒙 𝟐 +( 𝒚−𝟏) 𝟐 =𝟏𝟔. Graph the circle in the space provided.

Example 6c The standard equation of a circle is 𝒙 𝟐 +( 𝒚−𝟏) 𝟐 =𝟏𝟔. Graph the circle in the space provided. Center: (𝟎, 𝟏)

Example 6c The standard equation of a circle is 𝒙 𝟐 +( 𝒚−𝟏) 𝟐 =𝟏𝟔. Graph the circle in the space provided. Center: (𝟎, 𝟏) Radius: 𝒓= 𝟏𝟔 =𝟒

Example 6d The standard equation of a circle is (𝒙+𝟐) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟐𝟓. Graph the circle in the space provided.

Example 6d The standard equation of a circle is (𝒙+𝟐) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟐𝟓. Graph the circle in the space provided. Center: (−𝟐, 𝟑)

Example 6d The standard equation of a circle is (𝒙+𝟐) 𝟐 +( 𝒚−𝟑) 𝟐 =𝟐𝟓. Graph the circle in the space provided. Center: (−𝟐, 𝟑) Radius: 𝒓= 𝟐𝟓 =𝟓