1511 circle position-conics project

Introduction
A Closer Look at Graphing
Translations
Definition of a Circle
Definition of Radius
Definition of a Unit Circle
Deriving the Equation of a Circle
The Distance Formula
Moving The Center
Completing the Square
It is a Circle if?
Conic Movie
Conic Collage
End Next
Geometers Sketchpad: Cosmos
Geometers Sketchpad: Headlights
Projectile Animation
Planet Animation
Plane Intersecting a Cone Animation
Footnotes
Skip Intro

Next
Parabola Circle Ellipse Hyperbola
6
4
2
-2
-4
-10 -5
Quadratic Relations
PreviousMain MenuEnd

The quadratic relations that we studied in the beginning
of the year were in the form of y = Axy = Ax22
+ Dx + F+ Dx + F,
where A, D, and F stand for constants, and A ≠ 0.
This quadratic relation is a parabola, and it is the
only one that can be a function.
It does not have to be a function, though.
A parabola is determined by a plane intersecting a
cone and is therefore considered a conic section.
NextPreviousMain MenuEnd

The general equation for all conic sections is:
AxAx22
+ Bxy + Cy+ Bxy + Cy22
+ Dx + Ey + F = 0+ Dx + Ey + F = 0
where A, B, C, D, E and F represent constants
When an equation has a “yy22
” term and/or an “xyxy”
term it is a quadratic relation instead of a quadratic
function.
and where the equal sign could be replaced by an
inequality sign.

A Closer Look at Graphing Conics
Plot the graph of each relation. Select values of x and
calculate the corresponding values of y until there are
enough points to draw a smooth curve. Approximate all
radicals to the nearest tenth.
1)1) xx22
+ y+ y22
= 25= 25
2)2) xx22
+ y+ y22
+ 6x = 16+ 6x = 16
3)3) xx22
+ y+ y22
- 4y = 21- 4y = 21
4)4) xx22
+ y+ y22
+ 6x - 4y = 12+ 6x - 4y = 12
5)5) ConclusionsConclusions
NextPreviousMain MenuLast ViewedEnd

xx22
+ y+ y22
= 25= 25
x = 3
32
+ y2
= 25
9 + y2
= 25
y2
= 16
y2
=16
y =4
There are 22 points to graph: (3,4)
(3,-4)
y = ± 4
NextPreviousMain MenuA Close Look at GraphingEnd

Continue to solve in this manner, generating a table of values.
x y
-5 0
-4 ±3
-3 ±4
-2 ±4.6
-1 ±4.9
0 ±5
1 ±4.9
2 ±4.6
3 ±4
4 ±3
5 0
x y
-5 0
-4 ±3
-3 ±4
-2 ±4.6
-1 ±4.9
0 ±5
1 ±4.9
2 ±4.6
3 ±4
4 ±3
5 0

Graphing xx22
+ y+ y22
= 25= 25
x y
-5 0
-4 ±3
-3 ±4
-2 ±4.6
-1 ±4.9
0 ±5
1 ±4.9
2 ±4.6
3 ±4
4 ±3
5 0
x
y
-10 -5 5
6
4
2
-2
-4
-6

xx22
+ y+ y22
+ 6x = 16+ 6x = 16
x = 1
12
+ y2
+6(1) = 16
1 + y2
+6 = 16
y2
= 9
y2
=9
y =3
(1,-3)
y = ± 3

x y
-8 0
-7 ±3
-6 ±4
-5 ±4.6
-4 ±4.9
-3 ±5
-2 ±4.9
-1 ±4.6
0 ±4
1 ±3
2 0
x y
-8 0
-7 ±3
-6 ±4
-5 ±4.6
-4 ±4.9
-3 ±5
-2 ±4.9
-1 ±4.6
0 ±4
1 ±3
2 0

x y
-8 0
-7 ±3
-6 ±4
-5 ±4.6
-4 ±4.9
-3 ±5
-2 ±4.9
-1 ±4.6
0 ±4
1 ±3
2 0
Graphing xx22
+ y+ y22
+ 6x = 16+ 6x = 16
6
4
2
-2
-4
-6
-10 -5 5

xx22
+ y+ y22
- 4y = 21- 4y = 21
x = 3
32
+ y2
- 4y = 21
9 + y2
- 4y = 21
y2
- 4y -12 = 0
(3,-2)
y = 6 and y = -2
(y - 6)(y + 2) = 0
y - 6 = 0 and y + 2 = 0

0 -3, 7
3 -2, 6
4 -1, 5
5 2
1 -2.9, 6.9
2 -2.6, 6.6
x y
-5 2
-4 -1, 5
-3 -2, 6
-2 -2.6, 6.6
-1 -2.9, 6.9
0 -3, 7
3 -2, 6
4 -1, 5
5 2
1 -2.9, 6.9
2 -2.6, 6.6
x y
-5 2
-4 -1, 5
-3 -2, 6
-2 -2.6, 6.6
-1 -2.9, 6.9

Graphing xx22
+ y+ y22
- 4y = 21- 4y = 21
0 -3, 7
3 -2, 6
4 -1, 5
5 2
1 -2.9, 6.9
2 -2.6, 6.6
x y
-5 2
-4 -1, 5
-3 -2, 6
-2 -2.6, 6.6
-1 -2.9, 6.9
6
4
2
-2
-4
-6
-10 -5 5

xx22
+ y+ y22
+ 6x - 4y = 12+ 6x - 4y = 12
x = 1
12
+ y2
+ 6(1) - 4y = 12
1 + y2
+ 6 - 4y = 12
y2
- 4y - 5 = 0
(1,-1)
y = 5 and y = -1
(y - 5)(y + 1) = 0
y - 5 = 0 and y + 1 = 0

x y
-8 2
-7 -1, 5
-6 -2, 6
-5 -2.6, 6.6
-3 -3, 7
-2 -2.9, 6.9
-1 -2.6, 6.6
-4 -2.9, 6.9
0 -2, 6
1 -1, 5
2 2
x y
-8 2
-7 -1, 5
-6 -2, 6
-5 -2.6, 6.6
-3 -3, 7
-2 -2.9, 6.9
-1 -2.6, 6.6
-4 -2.9, 6.9
0 -2, 6
1 -1, 5
2 2

Graphing xx22
+ y+ y22
+ 6x - 4y = 21+ 6x - 4y = 21
x y
-8 2
-7 -1, 5
-6 -2, 6
-5 -2.6, 6.6
-3 -3, 7
-2 -2.9, 6.9
-1 -2.6, 6.6
-4 -2.9, 6.9
0 -2, 6
1 -1, 5
2 2
8
6
4
2
-2
-4
-10 -5 5

-10 -5 5 10
8
6
4
2
-2
-4
xx22
+ y+ y22
= 25= 25
xx22
+ y+ y22
+ 4x = 16+ 4x = 16
xx22
+ y+ y 22
- 6y = 21- 6y = 21
xx22
+ y+ y22
+ 4x - 6y = 12+ 4x - 6y = 12
What conclusions can you draw about the shape and location
of the graphs?

What conclusions can you draw about the shape and location
of the graphs?
• All of the graphs are circles. xx22
+ y+ y22
= 25= 25
10
5
-5
-10 10
• As you add an x-term, the graph moves left. xx22
+ y+ y22
+ 4x = 16+ 4x = 16
• As you subtract a y-term, the graph moves up. xx22
+ y+ y 22
- 6y = 21- 6y = 21
• You can move both left and up. xx22
+ y+ y22
+ 4x - 6y = 12+ 4x - 6y = 12

A circlecircle is the set of all points in a plane equidistant from a
fixed point called the centercenter.
-10 -5 5 10
6
4
2
-2
-4
-6
Center
CircleCircle
Center
GEOMETRICAL DEFINITION OF A
CIRCLE

A radiusradius is the segment whose endpoints are the center of
the circle, and any point on the circle.
6
4
2
-2
-4
-6
-10 -5 5 10
radiusradius
radius
radius

A unit circle is a circle with a radius of 1 whose center is at
the origin.
It is the circle on which all other circles are based.
-2 -1 1 2
1.5
1
0.5
-0.5
-1
the origin.
r =
1
r =
1
the origin.
Center: (0, 0)Center: (0, 0)
A unit circleunit circle is a circle with a radius of 1radius of 1 whose center is atcenter is at
the originthe origin.

The equation of a circle is derived from its radiusradius.
1
0.5
-0.5
-1
-1.5
-2 -1 1 2
NextPreviousMain MenuEnd Last Viewed

Use the distance formuladistance formula to find an equation for x and y.
This equation is also the equation for the circlethe equation for the circle.
NextPreviousMain Menu
1
0.5
-0.5
-1
-1.5
-2 -1 1 2
Deriving the Equation of a CircleEnd

THE DISTANCE FORMULA
(x2, y2)
(x1, y1)
D =(x2 −x1)2
+(y2 −y1)2
NextPreviousMain Menu
1
0.5
-0.5
-1
-1.5
-2 -1 1 2
Deriving the Equation of a CircleEnd

Deriving the Equation of a Circle
D =x2 −x1( )
2
+y2 −y1( )
2
r =x −0( )
2
+y −0( )
2
r =x2
+y2
r2
= x2
+y2
( )
2
1
0.5
-0.5
-1
-1.5
-2 -1 1
(x, y)(x, y)
(0, 0)(0, 0)
Let rr for radiusradius length replace D for distance.
r2
= x2
+ y2
rr22
= x= x22
+ y+ y22
Is the equation for a circle
with its center at the origin
and a radius of length r.

The unit circle therefore has the equation:
x2
+ y2
= 1
1
0.5
-0.5
-1
-1.5
-2 -1 1
(x, y)(x, y)
(0, 0)(0, 0)
r=
1
r=
1
NextPreviousMain MenuEnd Deriving the Equation of a Circle

x2
+ y2
= 4
If r = 2r = 2, then
1
0.5
-0.5
-1
-1.5
-2 -1 1
(x, y)(x, y)
(0, 0)(0, 0)
r=
2
r=
2
NextPreviousMain MenuDeriving the Equation of a CircleEnd

In order for a satellite to remain in a circular orbit above the
Earth, the satellite must be 35,000 km above the Earth.
Write an equation for the orbit of the satellite. Use the
center of the Earth as the origin and 6400 km for the radius
of the earth.
(x - 0)2
+ (y - 0)2
= (35000 + 6400)2
x2
+ y2
= 1,713,960,000
NextPreviousMain MenuDeriving the Equation of a CircleEnd

What will happen to the equation if the center is not at the
origin?
6
4
2
-2
-4
-6
-10 -5 5
r =x2 −x1( )
2
+y2 −y1( )
2
(1,2)(1,2)
r = 3
r = 3 (x,y)(x,y)
3 =x −1()
2
+y −2()
2
32
= x −1( )
2
+y −2( )
2





2
9 =x −1()
2
+y −2( )
2

-10 -5 5 10
No matter where the circle is located, or where the center
is, the equation is the same.
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
NextPreviousMain MenuMoving the CenterEnd

6
4
2
-2
-4
-10 -5 5 10
(x,y)(x,y)
Assume (x, y)(x, y) are the coordinates of a point on a circle.
(h,k)(h,k)
The center of the circle is (h, k)(h, k),
rr
and the radius is r.
Then the equation of a circle is: ((xx -- hh))22
+ (+ (yy -- kk))22
== rr22
.

((xx -- hh))22
+ (+ (yy -- kk))22
== rr22
((xx -- 00))22
+ (+ (yy -- 33))22
== 7722
(x)(x)22
+ (y - 3)+ (y - 3)22
= 49= 49
Write the equation of a circle with a center at (0, 3)center at (0, 3) and a
radius of 7radius of 7.
20
15
10
5
-5
-20 -10 10 20

Find the equation whose diameter has endpoints of
(-5, 2) and (3, 6).
First find the midpoint of the diameter using the
midpoint formula.
MIDPOINT
M(x,y) =
x1 +x2
2
,
y1 +y2
2






This will be the center.
M(x,y) =
−5 +3
2
,
2 +6
2






M(x,y) =−1, 4( )

(-5, 2) and (3, 6).
Then find the length distance between the midpoint
and one of the endpoints.
DISTANCE FORMULA
This will be the radius.
D =x2 −x1( )
2
+y2 −y1( )
2
D =−1−3( )
2
+4 −6()
2
D =−4()
2
+−2()
2
D =16+4
D =20

(-5, 2) and (3, 6).
Therefore the center is (-1, 4)
The radius is 20
(x - -1)2
+ (y - 4)2
= 20
2
(x + 1)2
+ (y - 4)2
= 20
-10
10
5
-5
NextPreviousMain MenuMoving the CenterSkip TangentsEnd

A line in the plane of a circle can intersectintersect the circle in 11 or
22 points. A line that intersects the circle in exactlyexactly oneone
point is said to be tangenttangent to the circle. The line and the
circle are considered tangent to each other at this point of
intersection.
Write an equation for a circle with center (-4, -3) that is
tangent to the x-axis.
-5
-2
-4
-6
A diagramdiagram will help.
(-4, 0)(-4, 0)
(-4, -3)(-4, -3)
33
A radius is always perpendicularperpendicular
to the tangent line.
(x + 4)2
+ (y + 3)2
= 9

The standard form equation for all conic sections is:
AxAx22
+ Bxy + Cy+ Bxy + Cy22
+ Dx + Ey + F = 0+ Dx + Ey + F = 0
where A, B, C, D, E and F represent constants and where the
equal sign could be replaced by an inequality sign.
How do you put a standard form equation into graphing form?
The transformation is accomplished through completing thecompleting the
squaresquare.

Graph the relation x2
+ y2
- 10x + 4y + 13 = 0.13
1. Move the FF term to the other side.
x2
+ y2
- 10x + 4y = -13
3. Complete the square for the x-termsx-terms and y-termsy-terms.
xx22
-- 10x10x ++ yy22
++ 4y4y = -13= -13
−10
2
=−5
−5()
2
=25
4
2
= 2
2()
2
=4
xx22
-- 10x + 2510x + 25 ++ yy22
++ 4y + 44y + 4 = -13= -13 + 25+ 25 + 4+ 4
(x - 5)(x - 5)22
++ (y + 2)(y + 2)22
== 1616
xx22
++ yy22
-- 10x10x ++ 4y4y ++ 1313 = 0= 0
xx22
++ yy22
-- 10x10x ++ 4y4y == -13-13
2. Group the x-termsx-terms and y-termsy-terms together
xx22
-- 10x10x ++ yy22
++ 4y4y == -13-13
Graph the relation
NextPreviousMain MenuCompleting the SquareEnd

(x - 5)(x - 5)22
++ (y + 2)(y + 2)22
== 1616
center: ((55,, -2-2))
radius = 44
(x - 5)(x - 5)22
++ (y + 2)(y + 2)22
== 4422
4
(5, -2)
-5 5 10
6
4
2
-2
-4
-6

What if the relation is an inequality?
xx22
+ y+ y22
- 10x + 4y + 13 < 0- 10x + 4y + 13 < 0
Do the same steps to transform it to graphing form.
(x - 5)(x - 5)22
+ (y + 2)+ (y + 2)22
< 4< 422
This means the values are inside the circle.
The values are less thanless than the radiusradius.
-5 5 10
6
4
2
-2
-4
-6

Write xx22
+ y+ y22
+ 6x - 2y - 54 = 0+ 6x - 2y - 54 = 0 in graphing form. Then
describe the transformation that can be applied to the
graph of x2
+ y2
= 64 to obtain the graph of the given
equation.
1. x2
+ y2
+ 6x - 2y = 54
2. x2
+ 6x + y2
- 2y = 54
3. (6
/2) = 3 (-2
/2) = -1
4. (3)2
= 9 (-1)2
= 1
5. x2
+ 6x + 9 + y2
- 2y + 1 = 54 + 9 + 1
6. (x + 3)2
+ (y - 1)2
= 64
7. (x + 3)2
+ (y - 1)2
= 82
8. center: (-3, 1) radius = 8

Write xx22
+ y+ y22
+ 6x - 2y - 54 = 0+ 6x - 2y - 54 = 0 in graphing form. Then
describe the transformation that can be applied to the
graph of xx22
+ y+ y22
= 64= 64 to obtain the graph of the given
equation.
xx22
+ y+ y22
= 64= 64 is translated 3 units left and one
unit up to become (x + 3)(x + 3)22
+ (y - 1)+ (y - 1)22
= 64= 64.
10
5
-5
-10
-20 -10 10 20

The graph of a quadratic relation will be a circlecircle if
the coefficientscoefficients of the xx22
term and yy22
term are
equalequal (and the xy term is zero).
PreviousMain MenuEnd

1511 circle position-conics project

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1511 circle position-conics project