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Writing Equations by Completing the Square Or Using the Distance Formula Circles Let’s start by reviewing the equation of a circle: (x – h)2 + (y – k)2 = r2 Write the equation of the circle with a center at (5, -6) and a radius of 7. (x – 5)2 + (y + 6)2 = 49 Find the center and radius of the circle give the equation. (x – 5)2 + (y - 3)2 = 4 Center (5, 3) r = 2 Circles Now we will find the equation by completing the square. You need to remember the standard equation of a circle and how to find “c”. (x – h)2 + (y – k)2 = r2 𝑏 2 c = ( ) 2 Circles Write the equation of the circle in standard form. x2 – 8x + y2 + 20y + 107 = 0 (x2 – 8x + ___) + (y2 + 20y + __) = - 107 +___ + ___ (x2 – 8x + 16) + (y2 + 20y + 100) = - 107+ 16 + 100 Group the x’s and the y’s and move the constant over. Don’t forget to put in the blanks. Find both of the c’s to fill in the blanks. (x – 4)2 + (y + 10)2 = 9 Center (4, - 10) r = 3 Factor the two equations and combine the numbers on the right. Now you have your equation!! Circles I know that seemed like a lot, so…. …try again! Transform the equation to standard form. x2 + y2 + 4x – 6y – 12 = 0 (x2 + 4x + ___) + (y2 - 6y + ___) = 12 + ___ +___ (x2 + 4x + 4) + (y2 - 6y + 9) = 12 + 4 + 9 (x + 2)2 + (y – 3)2 = 25 Center (-2, 3) r = 5 Circles Grab a white board and try a few on your own….. 1. x2 + y2 + 16x + 8y + 44 = 0 (x + 8)2 + (y + 4)2 = 36 Center (-8, -4) r = 6 2. x2 + y2 + 4x + 12y – 17 = 0 (x + 2)2 + (y + 6)2 = 57 Center (-2, -6) r = 57 3. x2 + y2 – 10x – 10y + 35 = 0 (x – 5)2 + (y – 5)2 = 15 Center (5, 5) r = 15 4. x2 + y2 + 2x – 8y + 5 = 0 (x + 1)2 + (y – 4)2 = 12 Center (-1, 4) r = 2 3 Circles Now we will use the distance formula to find an equation of a circle. If you have the center and a point, the distance from the center to that point will be the …… …..radius! These are the equations you will need: (x – h)2 + (y – k)2 = r2 d = (𝑥2 − 𝑥1 )2 +(𝑦2 − 𝑦1 )2 Circles The point (6, 8) lies on a circle centered at (2, 1). Find the equation of the circle in standard form. (x – 2)2 + (y – 1)2 = r2 r = (6 − 2)2 +(8 − 1)2 = 42 + 7 2 = 16 + 49 = 65 (x – 2)2 + (y – 1)2 = 65 The distance between the two points is the radius. ( 65)2 = 65 Circles Try a couple, they are pretty easy. Centered at (7, - 8) and passing through (10, -4) (x – 7)2 + (y + 8)2 = 25 Centered at (5, 6) and passing through (-1, -2) (x – 5)2 + (y – 6)2 = 100 Centered at (-4, -9) and passing through (1, 0) (x + 4)2 + (y + 9)2 = 106