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