conics can be formed by the intersection of a plane with a conical surface. if the plane passes...

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


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)


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


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


Write the equation of the parabola with focus ( 0 , -2 ) and directrix x = 3


Find the focus, directrix, vertex, and axis of symmetry. x2 + 4x + 2y + 10 = 0


Write the equation of the parabola with vertex ( 4 , 2 ) and directrix y = 5


Write the equation of the parabola with directrix y = 3 and focus ( 3 , 5 )


Find the focus, directrix, vertex, axis of symmetry, and length of the latus rectum. x2 – 4x – 12y – 32 = 0


Write this equation of a parabolain standard form:


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.


011161849 22 yxyx

Write the equation of the ellipse with center ( 0 , 0 ), a horizontal major axis, a = 6 and b = 4


Write the equation of the ellipse with x-intercepts and y-intercepts


2 3

Write the equation of the ellipse with foci ( -2 , 0 ) and ( 2 , 0 ), a = 7


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


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:


12516

22

yx

Find the center, foci, the length ofThe latus rectum, and graph thehyperbola:


1

9

2

25

)3( 22

xy

Find the center, foci, and vertices:


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.


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.


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.



Find the eccentricity for the following conic section: 4y2 – 8y + 9x2 – 54x + 49= 0


Find the eccentricity for the following conic section: 6y2 – 24y + 6x2 – 12= 0


Write the equation of the hyperbola with center ( -3 , 1 ) focus ( 2 , 1 ) and e = 5/4


Write the equation of an ellipse with center ( 0 , 3 ), major axis = 12, and eccentricity =2/3


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 )


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


Solve the following system of equations by Graphing.x2 = -4y5x2 + y2 = 25


Graph the following System, then statea sample solution.

3694

1622

22

yx

yx


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

conics can be formed by the intersection of a plane with a conical surface. if the plane passes...

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