Conics can be formed by the intersectionof a plane with a conical surface. If the plane passes through the Vertex of the conical surface, the intersection is aDegenerate Case (a point, a line, ortwo intersecting lines).
Conic Sections
There are 4 types of Conics which we will investigate:
1. Circles2. Parabolas3. Ellipses4. Hyperbolas
A circle is a set of points in the xy-plane that are afixed distance r from a fixed point (h, k). Thefixed distance r is called the radius, and the fixedpoint (h, k) is called the center of the circle.
x
y
(h, k)
r(x, y)
The standard form of an equation of acircle with radius r and center (h, k) is
x h y k r 2 2 2
Graph ( ) ( )x y 1 3 162 2 by hand.
( ) ( )x y 1 3 162 2
( ( )) ( )x y 1 3 42 2 2
( ) ( )x h y k r 2 2 2
h = -1, k = 3, r = 4
Center: (-1, 3), Radius: 4
(-1,3)
(3,3)
(-1, 7)
(-5, 3)
(-1, -1)
x
y
The general form of the equation ofa circle is
x y ax by c2 2 0
F i n d t h e c e n t e r a n d r a d i u s o fx y x y2 2 4 8 5 0 .
x x y y2 24 8 5
x x y y2 24 8 5 _ _
42
42
82
162
x x y y2 24 4 8 16 5 4 16
x y 2 4 252 2
Center: (2, -4), Radius: 5
Find the center, the radius and graph: x2 + y2 + 10x – 4y + 20 = 0
Find the equation of the circle with radius 3 in QI and tangent to the y-axis at ( 0 , 2 )
Find the equation of the circle with center at ( 2 , -1 ) through ( 5 , 3 )
Find the equation of the circle with endpoints of the diameter at ( 3, 5 ) and ( 3 , 1 )
Find the equation of the circle that Goes through these 3 points:(3, 4), (-1, 2), (0, 3)
A parabola is defined as the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. As a result, a parabola is the set of points P for which
d(F, P) = d(P, D)
The standard form of the equation of aparabola with directrix parallel to the y-axis is (opens left orright)
)(4)( 2 hxpky
The standard form of the equation of aparabola with directrix parallel to the x-axis is (opens up ordown) )(4)( 2 kyphx
Where (h, k) represents the vertex of the parabola and “p” represents the distance from the vertex to the focus.
The Axis of Symmetry is the line throughwhich the parabola is symmetrical.
The Latus Rectum is a line segment perpendicular to the Axis of Symmetrythrough the focus with endpoints on the parabola. The length of the LatusRectum is “4p”.
The Latus Rectum helps define the “width” of the parabola.
6
4
2
-2
-4
-5 5
Directrix
AXIS OF SYMMETRY
parabola
latus rectum
VERTEX
B
C
D
focus
Parabola with Axis of Symmetry Parallel to x-Axis, Opens to the Right, a > 0.
Equation Vertex Focus Directrix
y k a x h 2 4 (h, k) (h + a, k) x = -a + h
F = (h + a, k)
V = (h, k)
D: x = -a + hy
x
Axis of symmetry
y = k
Equation Vertex Focus Directrix
y k a x h 2 4 (h, k) (h - a, k) x = a + h
Parabola with Axis of Symmetry Parallel to x-Axis, Opens to the Left, a > 0.
D: x = a + h
F = (h - a, k)
Axis of symmetry y = k
y
x
V = (h, k)
Equation Vertex Focus Directrix
x h a y k 2 4 (h, k) (h, k + a) y = -a + k
Parabola with Axis of Symmetry Parallel to y-Axis, Opens up, a > 0.
D: y = - a + k
F = (h, k + a)
V = (h, k)
Axis of symmetry x = hy
x
Equation Vertex Focus Directrix
x h a y k 2 4 (h, k) (h, k - a) y = a + k
Parabola with Axis of Symmetry Parallel to y-Axis, Opens down, a > 0.
y
x
D: y = a + k
F = (h, k - a)
V = (h, k)
Axis of symmetry x = h
Find an equation of the parabola with vertex at the origin and focus (-2, 0). Graph the equation by hand and using a graphing utility.
Vertex: (0, 0); Focus: (-2, 0) = (-a, 0)
y ax2 4
y x2 4 2 ( )
y x2 8
The line segment joining the two points above and below the focus is called the latus rectum.
Let x = -2 (the x-coordinate of the focus)y x2 8y2 8 2 ( )
y2 16y 4
The points defining the latus rectum are (-2, -4) and (-2, 4).
10 0 10
10
10
(-2, -4)
(-2, 4)(0, 0)
Write the equation of a parabola with vertex ( 0 , 0 ) and focus ( 2, 0 )
Parabolas: Example Problems
Find the focus, directrix, vertex, and axis of symmetry. y2 – 12x – 2y + 25 = 0
Parabolas: Example Problems
Write the equation of the parabola with focus ( 0 , -2 ) and directrix x = 3
Parabolas: Example Problems
Find the focus, directrix, vertex, and axis of symmetry. x2 + 4x + 2y + 10 = 0
Parabolas: Example Problems
Write the equation of the parabola with vertex ( 4 , 2 ) and directrix y = 5
Parabolas: Example Problems
Write the equation of the parabola with directrix y = 3 and focus ( 3 , 5 )
Parabolas: Example Problems
Find the focus, directrix, vertex, axis of symmetry, and length of the latus rectum. x2 – 4x – 12y – 32 = 0
Parabolas: Example Problems
Write this equation of a parabolain standard form:
Parabolas: Example Problems
0632 yxy
Find the vertex, focus and directrix of
x x y2 4 8 20 0.
x x y2 4 8 20 0 x x y2 4 8 20
x x y2 4 8 20 _
42
42
x x y2 4 4 8 20 4 x y 2 8 32
x y 2 8 32
kyphx 42
Vertex: (h, k) = (-2, -3)
p = 2
Focus: (-2, -3 + 2) = (-2, -1)
Directrix: y = -2 + -3 = -5
10 0 10
10
10
(-2, -3)(-2, -1)
y = -5
(-6, -1)
(2, -1)
An ellipse is the collection of points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.
y
x
P = (x, y)
Focus2Focus1
Major Axis
Minor Axis
The standard form of the equation of anellipse with major axis parallel to the x-axis is
1)()(
2
2
2
2
b
ky
a
hx
The standard form of the equation of anellipse with major axis parallel to the y-axis is
1)()(
2
2
2
2
a
ky
b
hx
(h,k) is the center of the ellipse
For any ellipse,
“2a” represents the distance along themajor axis (a is always greater than b)
“2b” represents the distance along the minor axis
“c” represents the distance from the centerto either focus (the foci of an ellipse are always along the major axis)
222 cba
Ellipse with Major Axis Parallel to the x-Axis where a > b and b2 = a2 - c2.
Equation Center Foci Vertices
x h
a
y k
b
2
2
2
21
(h, k) (h + c, k) (h + a, k)
(h - a, k) (h + a, k)(h, k)
(h - c, k) (h + c, k)y
x
Major axis
Ellipse with Major Axis Parallel to the x-Axis
(h, k)
Focus 1 Focus 2y
x
Major axis
The ellipse is like a circle, stretched morein the “x” direction
Ellipse with Major Axis Parallel to the y-Axis where a > b and b2 = a2 - c2.
Equation Center Foci Vertices
x h
b
y k
a
2
2
2
21
(h, k) (h, k + c) (h, k + a)
y
x
(h, k + a)
(h, k - a)
(h, k)
(h, k + c)
(h, k - c)
Major axis
Ellipse with Major Axis Parallel to the y-Axis
y
x
(h, k)
Focus 1
Focus 2
Major axis
The ellipse is like a circle, stretched morein the “y” direction
Sketch the ellipse and find the center, foci, and the length of the major and minor axes:
Ellipses: Example Problems
1
16
5
25
4 22
yx
Find the center and the foci. Sketch the graph.
Ellipses: Example Problems
011161849 22 yxyx
Write the equation of the ellipse with center ( 0 , 0 ), a horizontal major axis, a = 6 and b = 4
Ellipses: Example Problems
Write the equation of the ellipse with x-intercepts and y-intercepts
Ellipses: Example Problems
2 3
Write the equation of the ellipse with foci ( -2 , 0 ) and ( 2 , 0 ), a = 7
Ellipses: Example Problems
Write the equation of this ellipse
Ellipses: Example Problems6
4
2
-2
-4
-6
-5 5
Find the center, foci, and graph theellipse: 16x2 + 4y2 – 96x + 8y + 84 = 0
Ellipses: Example Problems
The length of the Latus Rectum for anEllipse is
By knowing the Latus Rectum, it makes the graph
of the ellipse more accurate
Ellipses: Latus Rectum
a
b22
(h, k)
Latus RectumLatus
Rectum
Use the length of the latus rectum in Graphing the following ellipse:
Ellipses: Latus Rectum
116
)1(
36
)5( 22
yx
Find an equation of the ellipse with center at the origin, one focus at (0, 5), and a vertex at (0, -7). Graph the equation by hand
Center: (0, 0)
Major axis is the y-axis, so equation is of the form
x
b
y
a
2
2
2
2 1
Distance from center to focus is 5, so c = 5
Distance from center to vertex is 7, so a = 7
b a c2 2 2 2 27 5 49 25 24
x
b
y
a
2
2
2
2 1
x y2 2
224 71
x y2 2
24 491
and a b 7 242,
5 0 5
5
5
(0, 7)
(0, -7)
FOCI
( 24,0) ( 24,0)
Find the center, m ajor ax is, foci, and vertices of4 9 32 36 64 02 2x y x y
4 32 9 36 642 2x x y y
4 8 9 4 642 2x x y y _ _
82
162
42
42
4 8 16 9 4 4 64 64 362 2x x y y
4 4 9 2 362 2x y
x y
4
9
2
41
2 2
x y
49
24
12 2
x h
a
y k
b
2
2
2
2 1
Center: (h, k) = (-4, 2)
Major axis parallel to the x-axis
c a b2 2 2 9 4 5
Vertices: (h + a, k) = (-4 + 3, 2) or (-7, 2) and (-1, 2)
Foci: (h + c, k) = ( , )
, ) ( ,
4 5 2
4 5 4 5
or
( 2 and 2)
8 6 4 2 0
4
2
2
4
V(-7, 2) V(-1, 2)
C (-4, 2)
F(-6.2, 2) F(-1.8, 2)
(-4, 4)
(-4, 0)
A hyperbola is the collection of points in the plane the difference of whose distances from two fixed points, called the foci, is a constant.
The standard form of the equation of ahyperbola with transverse axis parallel to x-axis is
1)()(
2
2
2
2
b
ky
a
hx
The standard form of the equation of ahyperbola with transverse axis parallel to y-axis is
1)()(
2
2
2
2
b
hx
a
ky
(h,k) is the center of the hyperbola
For any hyperbola,
“2a” represents the distance along thetransverse axis
“2b” represents the distance along the conjugate axis
“c” represents the distance from the centerto either focus (the foci of a hyperbola are always along the transverse axis)
222 bac
The length of the Latus Rectum for aHyperbola is
a
b22
The equations of the asymptotes for the hyperbola arethese if there is a Horizontal Transverse Axis
or these if there is a VerticalTransverse Axis
)()( hxa
bky
)()( hxb
aky
Hyperbola with Transverse Axis Parallel to the x-Axis
Latus Rectum
Hyperbola with Transverse Axis Parallel to the y-Axis Latus Rectum
Find an equation of a hyperbola with center at the origin, one focus at (0, 5) and one vertex at (0, -3). Determine the oblique asymptotes. Graph the equation by hand and using a graphing utility.
Center: (0, 0) Focus: (0, 5) = (0, c)
Vertex: (0, -3) = (0, -a)
Transverse axis is the y-axis, thus equation is of the form
y
a
x
b
2
2
2
2 1
y
a
x
b
2
2
2
2 1 a c2 29 25 ,
b c a2 2 2 = 25 - 9 = 16
y x2 2
9 161
Asymptotes: yab
x x 34
5 0 5
5
5
V (0, 3)
V (0, -3)
(4, 0)(-4, 0)
F(0, 5)
F(0, -5)
y x34
y x 34
y x2 2
9 161
Find the center, transverse axis, vertices, foci, and asymptotes of 4 16 8 16 02 2x x y y .
4 16 8 16 02 2x x y y
4 4 8 162 2x x y y
4 4 8 162 2x x y y _ _
42
42
82
162
4 4 4 8 16 16 16 162 2x x y y
4 2 4 162 2x y
x y
2
4
4
161
2 2
x y
24
416
12 2 x h
a
y k
b
2
2
2
2 1
Center: (h, k) = (-2, 4)
Transverse axis parallel to x-axis.
a b c a b2 2 2 2 24 16 4 16 20 , ,
a b c 2 4 2 5, , Vertices: (h + a, k) = (-2 + 2, 4) or (-4, 4) and (0, 4)
Foci: or
and
( , ) ( , )
( , ) ( , )
h c k
2 2 5 4
2 2 5 4 2 2 5 4
Asymptotes: y kba
x h
y x 442
2( )
y x 4 2 2
a b c 2 4 2 5, , (h, k) = (-2, 4)
10 0 10
10
C(-2,4)
V (-4, 4) V (0, 4)
F (2.47, 4)F (-6.47, 4)
(-2, 8)
(-2, 0)
y - 4 = -2(x + 2) y - 4 = 2(x + 2)
Write the equation of the hyperbola with center ( 4 , -2 ) a focus ( 7 , -2 ) and a vertex ( 6, -2 )
Hyperbolas: Example Problems
Find the center, foci, and graph the hyperbola:
Hyperbolas: Example Problems
12516
22
yx
Find the center, foci, the length ofThe latus rectum, and graph thehyperbola:
Hyperbolas: Example Problems
1
9
2
25
)3( 22
xy
Find the center, foci, and vertices:
Hyperbolas: Example Problems
16x2 – 4y2 – 96x + 8y + 76 = 0
Equilateral HyperbolasEquilateral Hyperbola: A hyperbola where a = b.
When we have an equilateral hyperbolawhose asymptotes are the coordinate axes, the equation of the hyperbola lookslike this: xy = k.This type of hyperbola is called a rectangular hyperbola, and is easier to graph because the asymptotes are the x and y axes.
Rectangular Hyperbolas6
4
2
-2
-4
-6
-5 5
6
4
2
-2
-4
-6
-5 5
The equation of an rectangular hyperbolais xy = k (where k is a constant value).
If k >0, then your graph looks like this:
If k<0, then your graph looks like this:
Rectangular HyperbolasExample: Graph by hand the hyperbola: xy = 6.
6
4
2
-2
-4
-6
-5 5
The General form of the equation of anyconic section is…
022 FEyDxCyBxyAxWhere A, B, and C are not all zero (however, for all of the examples we have studied so far, B = 0).
If A = C, then the conic is a…
If either A or C is zero, then we have a…
If A and C have the same sign, but A does not equal C, then the conic is a…
If A and C have opposite signs, then we have a ….
Let D denote a fixed line called the directrix; let F denote a fixed point called the focus, which is not on D; and let e be a fixed positive number called the eccentricity. A conic is the set of points P in the plane such that the ratio of the distance from F to P to the distance from D to P equals e. Thus, a conic is the collection of points P for which
d F Pd D P
e,,
Conic Sections: Eccentricity
To each conic section (ellipse, parabola, hyperbola, circle) there is a number called the eccentricity that uniquely characterizes the shape of the curve.
Conic Sections: Eccentricity
If e = 1, the conic is a parabola.
If e = 0, the conic is a circle.
If e < 1, the conic is an ellipse.
If e > 1, the conic is a hyperbola.
Conic Sections: Eccentricity
For both an ellipse and a hyperbola
eca
where c is the distance from the center to the focus and a is the distance from the center to a vertex.
Conic Sections: Eccentricity
Conic Sections: Eccentricity
Find the eccentricity for the following conic section: 4y2 – 8y + 9x2 – 54x + 49= 0
Conic Sections: Eccentricity
Find the eccentricity for the following conic section: 6y2 – 24y + 6x2 – 12= 0
Conic Sections: Eccentricity
Write the equation of the hyperbola with center ( -3 , 1 ) focus ( 2 , 1 ) and e = 5/4
Conic Sections: Eccentricity
Write the equation of an ellipse with center ( 0 , 3 ), major axis = 12, and eccentricity =2/3
Conic Sections: Eccentricity
Write the equation of the ellipse and find the eccentricity, given it has foci ( 1 , -1 ) and ( 1 , 5 ) and goes through the point ( 4, 2 )
Conic Sections: Eccentricity
Find the center, the foci, and eccentricity. EX 1: 4x2 + 9y2 = 36
EX 2: 4y2 – 8y - 9x2 – 54x + 49 = 0 EX 3: 25x2 + y2 – 100x + 6y + 84 = 0
Conic Sections: Solving Systems of Equations Graphically
Solve the following System of Equations by Graphing.9x2 + 9y2 = 36Y – 4x = 5
Conic Sections: Solving Systems of Equations Graphically
Solve the following system of equations by Graphing.x2 = -4y5x2 + y2 = 25
Conic Sections: Solving Systems of Equations Graphically
Graph the following System, then statea sample solution.
3694
1622
22
yx
yx
Conic Sections: Solving Systems of Equations Graphically
Graph the following System, then statea sample solution.
4
364)1(9 22
xy
yx
Theorem Identifying Conics without Completing the Square
Excluding degenerate cases, the equation
Ax Cy Dx Ey F2 2 0
where either or A C 0 0:
(a) Defines a parabola if AC = 0.
(b) Defines an ellipse (or a circle) if AC > 0.
(c) Defines a hyperbola if AC < 0.
Identify the equation without completing the square.
3 4 9 10 02 2x y x y A C 3 4,
AC 12 0The equation is a hyperbola.
The standard form of an equation of a circle of radius r with center at the origin (0, 0) is
x y r2 2 2