Conic Sections

Circles Ellipses

Parabolas Hyperbolas

Systems

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Day 1Day 2

Day 1Day 2

CirclesDefinitions

1) A circle is the set of all points, in a plane, equidistant from a fixed point.

2) A circle is the intersection of a right circular cone and a plane perpendicular to the axis of the cone.

Standard form: 222 )()( rkyhx

),( kh

022 FEyDxCyAx

Center:

Radius: (h, k)

rr

General form: where 0 and 0 and CACA

36)2()1( 22 yx

1)7( 22 yx

24)1()3( 22 yx

Find the center, radius

Center:

r =

5022 yx

Center:

r =

r = Center:

r = Center:

and graph.

Write the equation of the circle

Center: (2, -1) and r = 8Center: (-3, 0) and r = 3The center is at (2, -4) and the circle is tangent to the x-axisThe diameter of the circle has endpoints at (2, 6) and (8, -2)The center is at (4, -2) and the circle passes through the point (5, 3)

Write in standard form by completing the square

0114222 yxyx 010222 xyx 2 22 2 4 8 4x y x y 03061022 yxyx

EllipsesDefinitions1) An ellipse is the set of all points, in a plane, such that the sum of the

distances of each point from two fixed points is a constant

2) An ellipse is the intersection of a right circular cone and a plane not perpendicular to the axis of the cone.

Vocabulary: Foci - Each fixed point is called a focus of the ellipse.

Center: the midpoint of the line segment joining the foci and the ellipse

Major Axis: a line segment with endpoints on the ellipse and containing the foci

Minor Axis: line segment with endpoints on the ellipse and perpendicular to the major axis at the center of the ellipse F FC

F

F

C

General form: where 0 and 0 and CACA022 FEyDxCyAx

2 2 2c a b

Horizontal Vertical

Standard Equation

Center

Length of Major Axis

Length of minor axis

How to find c

Length of Focal Chord

Foci located at

2 2

2 21

x h y k

a b

2 2

2 21

x h y k

b a

2 2 2c a b

(h, k) (h, k)

2aThe major axis is horizontal

2aThe major axis is vertical

2bThe minor axis is vertical

2bThe minor axis is horizontal

2c 2c

(h - c, k) and (h + c, k) (h, k - c) and (h, k + c)

Find the center, foci, length of major and minor axes and graph.

22 21

9 4

yx

2 21 3

14 25

x y

Center:

Foci:

Length of Major axis

Length of minor axis

22 1( 1)1

5 20

yx

Write an equation for the graph.

Center:

Foci:

Length of Major axis

Length of minor axis

Write in standard form by completing the square

Ellipses

2 26 2 12 23 0x y x y 2 212 5 60x y 2 24 9 40 18 73 0x y x y 2 216 4 32 24 12 0x y x y

Write the equation of the ellipse in standard form.

The major axis is 16 units long and parallel to the x-axis. The center is at (5, 4) and minor axis is 9 units long.

The foci are at (0,-3) and (0,3). The length of the minor axis is 4.The vertices are at (-11, 5) and (7,5) and the minor axis is 4 units long.

Parabolas

Definitions1) A parabola is the intersection of right circular cone

and a plane parallel to an element of the cone

2) A parabola is the set of points in a plane each of which is the same distance from a fixed point as it is from a fixed line.

Vocabulary:

Focus - the fixed point (always located inside the parabola)

FV

F

VDirectrix - the fixed line (never intersects the parabola)

Vertex - the point at which the axis intersects the parabola

Axis of Symmetry - a line drawn through the focus, perpendicular to the directrix (sometimes called the axis of the parabola)

Focal Chord - any segment joining two points on the parabola and passing through the focus

Focal Radius - any segment joining the focus to a point on the parabola

Latus Rectum - the focal chord which is perpendicular to the axis and contains the focus.

Horizontal Vertical

Standard Equation

Direction of opening

Vertex

Axis ofSymmetry

Location of focus

Directrix

Length ofLatus Rectum

21

4y k x h

c

If > 0 opens up 1

4c

If < 0 opens down1

4c(h, k)

(h, k + c)

y = k - c

|4c|

21

4x h y k

c

If > 0 opens right 1

4c

If < 0 opens left1

4c(h, k)

(h + c, k)

x = h - c

x = hy = k

|4c|

General form: Where but not both 0 or 0A C 2 2 0Ax Cy Dx Ey F

Graph then find the requested information.

Vertex:

Focus:

(Length of latus rectum = )

Endpoints of latus rectum:

Axis of symmetry

Equation of directrix

216 3

8y x 21

5 18

x y 215 4

4y x 21

4 212

x y

Write the equation of the parabola in standard form.

2 16 8 80 0x x y 23 8 31x y y 22 4 8 18 0y y x

Write the equation of the parabola in standard form.

The focus is at (3, 8) and the directrix is y = 4.The vertex is (-7, 4), the length of the latus rectum is 6 and the graph opens left.The directrix is y = 2 and the right endpoint of the latus rectum is (6, -2)

Hyperbolas

Definitions

1) An hyperbola is a set of points in a plane such that for each, the absolute value of the difference of its distances from the two fixed points is a constant.

2) An hyperbola is the intersection of a right circular cone and a plane cutting both nappes of the cone.

Vocabulary

Foci - the two fixed points F F

F

F

Center - the midpoint of the transverse axis

C

C

Transverse Axis - a segment of the line passing through the foci with vertices as endpoints

Vertices - points of intersection of the branches of the hyperbola and the transverse axis

V

V

V

V

Conjugate Axis - perpendicular to the transverse axis at the center

Asymptote - a straight line approached by a given curve as one of the variables in the equation of the curve approaches infinity

General form: Where 2 2 0Ax Cy Dx Ey F 0 and 0 and 0C A C

Horizontal Vertical

Standard Equation

Center

Vertices

Length of Transverse Axis

Length of Conjugate Axis

How to find c

Foci located at

Equation of Asymptotes

2 2

2 21

x h y k

a b

(h, k)

(h – a, k) and (h + a, k)

2a

2b

c2 = a2 + b2

(h – c, k) and (h + c, k)

by k x h

a

2 2

2 21

y k x h

a b

(h, k)

(h, k – a) and (h, k + a)

2a

2b

c2 = a2 + b2

(h, k - c) and (h, k + c)

ay k x h

b

Graph the hyperbola and find the requested information.

Center:

Foci:

Vertices:

Equation of asymptotes:

2 2

125 36

x y 2 21 ( 2)

116 9

y x

2 2( 5)1

25 9

y x 1

36

)1(

36

)2( 22

yx

Write an equation for the graph.

Write the equation of the hyperbola in standard form.

036122434 22 yxyx 0968254 22 xyx 03624100425 22 yxyx

Write the equation of the hyperbola in standard form.

A vertex is (-4, 4) and the equation of the asymptotes is The center is (-3, 2), a vertex is (-3, 5) and one endpoint of the conjugate axis is -8, 2)A vertex is (4, 0) and the foci are at (6, 0) and (-6,0) 4

4

31 xy

2 23 4 8 8 0x y y 2 26 6 24 12 11 0x y x y 22 4 3 8 0x x y 2 25 5 10 2 0x y y

Ellipse

Hyperbola

Parabola

Circle

Identify the type of Conic section

Identify the type of conics, sketch a graph and solve the system.

1

2522

xy

yx

xy

yx

1436

22

25

722

22

yx

yx

5

2)3( 2

xy

yx

6464

6422

22

yx

yx

xy

yx

7

922