midpoint formula: distance formula: (x 1, y 1 ) (x 2, y 2 ) 1)(- 3, 2) and (7, - 8) 2)(2, 5) and (4,...
TRANSCRIPT
• 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.