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Circles

Circles. 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 = r 2 Write the equation of the circle with a center at (5, -6) and a radius of 7. (x – 5) 2 + (y + 6) 2 = 49

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Circles

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  1. Circles Writing Equations by Completing the Square Or Using the Distance Formula

  2. 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

  3. 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 • c =

  4. 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 • (x – 4)2 + (y + 10)2 = 9 • Center (4, - 10) r = 3 • 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. • Factor the two equations and combine the numbers on the right. • Now you have your equation!!

  5. 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

  6. 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 = • 3. x2 + y2 – 10x – 10y + 35 = 0 • (x – 5)2 + (y – 5)2 = 15 Center (5, 5) r = • 4. x2 + y2 + 2x – 8y + 5 = 0 • (x + 1)2 + (y – 4)2 = 12 Center (-1, 4) r = 2

  7. 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 =

  8. 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 = • = • = • = • (x – 2)2 + (y – 1)2 = 65 • The distance between the two points is the radius.

  9. 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

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