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Conic Sections
Circles Ellipses
Parabolas Hyperbolas
Systems
Day 1Day 2
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