Circles

Writing Equations by Completing the Square

Or Using the Distance Formula

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

c =

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

Circles

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!!

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 = 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

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 =

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

Circles

The distance between the two points is the radius.

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

Circles