• Midpoint formula:

2,

22121 yyxx

221

221 )()( yyxxd

• Distance formula:

(x1, y1)

(x2, y2)

1) (- 3, 2) and (7, - 8)

2) (2, 5) and (4, 10)

1) (1, 2) and (4, 6)

2) (-2, -5) and (3, 7)

COORDINATE PLANE FORMULAS:

3,22

82,

2

73

2

15,3

2

510,

2

42

22 )62()41(5

22 )75()32(13

CIRCLE: The set of all points that are equidistant from a given point.

GIVEN POINT: CENTER

EQUIDISTANT: RADIUS

(x1, y1)

(x2, y2)

(x3, y3)

d1

d2

d3

(x, y)

21

211 )()( yyxxd

Distance #1: (x1, y1)

22

222 )()( yyxxd

Distance #2 : (x2, y2)

23

233 )()( yyxxd

Distance #3: (x3, y3)

If all 3 points are on the circle, then all distances are equal!!

d1 = d2 = d3

CIRCLE FORMULA: Standard Form222 )()( rkyhx

Center: (h, k)

Radius: r(x, y)

(h, k)

222

22

221

221

)()(

)()(

)()(

ykxhr

ykxhr

yyxxd

Derive Formula: Distance

1. IDENTIFY the center and radius in the equation.

a.

Center: _________ Radius: ________

b.

Center: _________ Radius: ________

c.

Center: _________ Radius: ________

12)1()3( 22 xy

16)5()2( 22 yx

81)7()4( 22 yx

PRACTICE #1: Interpret Equation of a Circle

4

162

r

r(2, -5)

(4, 7) 9

812

r

r

(-1, -3) 32

12

122

r

r

r

2. Write an equation of the circle with a center (-1, 3) and radius of 6.

PRACTICE #2: Write the Equation of a Circle

3. Write the equation of the circle pictured to the right

22 36)6( r

36)3()1( 22 yx

22 9)3( r

9)2()2( 22 yx

4. (-1, 7) and (5, -1)

PRACTICE #3: Write the equation of the circle given the endpoints of a diameter.

5. (-3, 4) and (-7, -6)

Center: (2, 3) Center: (-5, -1)

222 25)37()21( r

222 )3()2( ryx

29)1()5( 22 yx

222 )1()5( ryx 222 29)14()53( r

25)3()3( 22 yx

6. (-3, -5) and (6, 2)

PRACTICE #3 : Continued7. (4, 8) and (4, -2)

Center: (4, 3)

222 )3()4( ryx 222 25)38()44( r

25)3()4( 22 yx

Center: (1.5, -1.5)

222 )2

3()

2

3( ryx

222

2

65)

2

36()

2

33( r

2652

232

23 )()( yx

HOW TO: Writing Circles in standard form

16)4(;428 2

361629)3612()168( 22 yyxx

02912822 yxyx

Center: (-4, 6) Radius: 9

Step #1: Group x and y terms separately together

Step #2: Move the constant term to the opposite side

Step #3: Complete the square for x’s and y’s (Add Both to Right Side)

81)6()4( 22 yx

029128 22 yyxxStep #1:

Step #2: 29128 22 yyxx

Step #3: ______29___)12(___)8( 22 yyxx

36)6(;6212 2

PRACTICE #4: Writing Circles in Standard Form

[A]

Write in standard form, find the radius and center. Sketch a graph

[B]058422 yxyx 07622 xyx

16)3(

37)36(22

2222

yx

yxx

Center: (3, 0)

Radius: r = 4

27)4()2(

1647)168()44(

7)8()4(

22

22

22

yx

yyxx

yyxx

Center: (2, -4)

Radius: 33r

PRACTICE #4: Continued

[C]

Write in standard form, find the radius and center. Sketch a Graph.

[D]0310622 yxyx 04

7322 xyx

Center: ( -3/2, 0)

Radius: r = 2

4)()2

3(

4

9

4

7)()

4

93(

4

7)()3(

22

22

22

yx

yxx

yxx2593)2510()96(

3)10()6(22

22

yyxx

yyxx

Center: (3, -5)

Radius: 37

37)5()3( 22 yx

098622 yxyx

9)8()6( 22 yyxx

[E]

Center: (-3, -4)

Radius: r = 4

16)4()3( 22 yx

PRACTICE #4: Continued Write in standard form, find the radius and center.

064161022 yxyx[F]

1699)168()96( 22 yyxx 642564)6416()2510( 22 yyxx

64)16()10( 22 yyxx

25)8()5( 22 yx

Center: (5, -8)

Radius: r = 5

PRACTICE #5: Equations given the a Tangent

[A] Center: (-4, -3) Tangent to x-axis

Write the equation of the circle given its tangency to an axis.

[B] Center: (3, 5) tangent to y-axis

9)5()3( 22 yx

TANGENT: A line intersecting at exactly one point with another curve.

303 r

6)3()4( 22 yx

3)3(0 r

Additional Fact: Tangents are perpendicular to the curve.