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Apr 4­11:19 AM

Welcome to the world of conic sections!

Some examples of conics in the real world:

http://www.youtube.com/watch?v=bFOnicn4bbg

Parabolas Ellipse

Circle

Hyperbola

Your Assignment:-Find at least four pictures (differentfrom the ones above) of conic sections inthe real world. You must have at least one picture of each type of conic section.-Identify what each pictures represents -parabola, hyperbola, circle or ellipse.-Glue to 8 x 10 in. or 9 x 12 in. paper.-Write your name on the front or back.-Be creative!-Due Tuesday, April 17th

Apr 4­10:43 AM

Chapter 10: Conic Sections

Circles

Standard Form of the Equation of a Circle: (x - h)2 + (y - k)2 = r2

r is the __________ and (h,k) is the ____________.

Examples:

1.) Write the equation of a circle in standard form with radius 3 and center (-3,4). Then graph the circle.

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Apr 4­10:57 AM

2.) Write the equation of the circle in standard form whose center is (6,1) and is tangent to the y-axis. Then graph the circle.

General Form of the Equation of a Circle: x2 + y2 + Dx + Ey + F = 0D, E and F are ___________.

Midpoint of a Line Segment: If the coordinates of P1 and P2 are (x1,y1) and (x2,y2), respectively, then the midpoint of P1P2 hascoordinates (x1 + x2, y1 + y2).

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Apr 4­10:59 AM

Review: Completing the square

1.) x2 + 4x - 5 = 0 2.) 2x2 + 4x - 8 = 0

Examples: Write the equation of the circle in standard form. Then graph each circle.

1.) x2 + y2 - 4x + 12y + 30 = 0

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Apr 4­11:03 AM

2.) 2x2 + 2y2 - 4x + 12y - 18 = 0

Distance Formula for Two Points:The distance between two points (x1,y1) and (x2,y2) in the coordinate plane is given by d = √(x2 - x1)2 + (y2 - y1)2 .

Example: Find the distance between the points (-1,-6) and (5,-3).

Apr 4­11:16 AM

Examples: Write the equation of the circle that satisfies each set of conditions.

1.) The circle passes through the point (5,6) and has its centerat (-4,3).

2.) The endpoints of a diameter are at (2,3) and at (-6,-5).

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Apr 4­11:18 AM

Ellipses

-Ellipse:

-Foci:

-Center:

-Minor Axis:

-Major Axis:

-Vertices:

CF1 F2

A

B

D

E

Apr 4­11:56 AM

Standard Form ofthe Equation of Orientation Descriptionand Ellipse

(x - h)2 + (y - k)2 = 1, -Center: (h,k) a2 b2 -Foci: (h±c, k)where c2 = a2 - b2. -Major Axis: y = k

-Major Axis Vertices: (h±a, k)-Minor Axis: x = h-Minor Axis Vertices: (h, k±b)

(y - k)2 + (x - h)2 = 1, -Center: (h,k) a2 b2 -Foci: (h, k±c)where c2 = a2 - b2. -Major Axis: x = h

-Major AxisVertices: (h, k±a)-Minor Axis: y = k-Major AxisVertices: (h±b, k)

x = h

y = k(h,k)

x = h

y = k(h,k)

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Apr 5­9:09 AM

Examples:

1.) Consider the ellipse graphed at the right.

a.) Write the equation of the ellipse in standard form.

b.) Find the coordinates of the foci.

(2,­4) (8,­4)

(2,­7)

Apr 5­9:15 AM

2.) For the equation (y - 3)2 + (x + 4)2 = 1, find the coordinates 25 9

of the center, foci and vertices of the ellipse. Then graph.

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Apr 5­9:17 AM

3.) Find the coordinates of the center, the foci and the vertices of the ellipse with the equation 4x2 + 9y2 - 40x + 36y + 100 = 0.Then graph the ellipse.

Apr 5­9:21 AM

Hyperbolas

-Hyperbola:

-Foci:

-Center:

-Vertex:

-Asymptotes:

-Transverse Axis:

-Conjugate Axis:

F1F2

vertices

asymptot

easymptote center

conjugate axis

transverse axis

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Apr 5­10:22 AM

Standard Form ofthe Equation of a Orientation DescriptionHyperbola

(x - h)2 - (y - k)2 = 1, Center: (h,k) a2 b2 Foci: (h±c, k)where b2 = c2 - a2. Vertices: (h±a, k)

Equation of transverse axis: y = k (parallel tox-axis)Asymptotes: y - k = ±(b/a)(x-h)

(y - k)2 - (x - h)2 = 1, Center: (h, k) a2 b2 Foci: (h, k±c)where b2 = c2 - a2. Vertices: (h, k±a)

Equation of transverseaxis: x = h (parallel toy-axis)Asymptotes: y - k = ±(a/b)(x-h)

x = h

y = k

x = h

y = k

(h,k)

(h,k)

Apr 5­10:45 AM

Examples:

1.) Find the coordinates of the center, the foci, the vertices and the equations of the asymptotes of the hyperbola whose equation is x2 - y2 = 1. Then graph. 25 4

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Apr 5­11:17 AM

2.) Find the coordinates of the center, foci, vertices and the equations of the asymptotes of the graph of 9x2 - 4y2 - 54x - 40y - 55 = 0. Then graph.

Apr 5­11:21 AM

Parabolas

-Parabola:

-Focus:

-Directrix:

-Axis of symmetry:

-Vertex:

directrix

vertex

focusaxis of symmetry

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Apr 5­11:45 AM

Standard Form of theEquation of a Parabola Orientation Description

(y - k)2 = 4p(x - h) Vertex: (h, k)Focus: (h + p, k)Axis of symmetry: y = kDirectrix: x = h - pOpening: Right if p > 0

Left if p < 0

(x - h)2 = 4p(y - k) Vertex: (h, k)Focus: (h, k + p)Axis of symmetry: x = hDirectrix: y = k - pOpening: Upward if p > 0

Downward if p < 0

x = h - p

y = k(h + p, k)

(h, k)

(h, k)

(h, k + p

)

y = k - p

x = h

Apr 5­12:02 PM

Examples: For the equation of each parabola, find the coordinates of the vertex and focus and the equations of the directrix and axis of symmetry. Then graph.

1.) x2 = 12(y - 1)

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Apr 5­12:29 PM

2.) y2 - 4x + 2y + 5 = 0

Apr 5­12:31 PM

Examples: Write the equation of the parabola that meets each set of conditions. Then graph.

1.) The vertex is at (-5,1) and the focus is at (2,1).

2.) The axis of symmetry is y = 6, the focus is at (0,6) and p = -3.

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Apr 5­12:37 PM

Chapter 10 Homework Name: ______________

1.) Write the standard form of the equation of the circle with center (4,-7) and radius √3. Then graph.

2.) Write the standard form of the equation of the circlegraphed below.

Apr 5­12:48 PM

3.) Write the standard form of the equation of the circle. Then graph. 6x2 - 12x + 6y2 + 36y = 36

4.) Write the equation of the circle whose endpoints of a diameter are (-3,4) and (2,1).

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Apr 5­12:57 PM

For each equation of the ellipse, find the coordinates of the center, foci and vertices. Then graph each equation.

5.) x2 + (y - 4)2 = 1 81 49

6.) 9x2 + 4y2 - 18x + 16y = 11

Apr 5­1:01 PM

Write the equation of each ellipse in standard form. Then find the coordinates of the foci.7.)

8.)

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Apr 5­1:08 PM

9.) Determine which of the following equations matches the graph of the hyperbola below.

A.) x2 - y2 = 14

B.) y2 - x2 = 14

C.) x2 - y2 = 14

D.) y2 - x2 = 1 4

10.) Write the equation of a hyperbola centered at the origin,with a = 8, b = 5 and transverse axis on the y-axis.

Apr 5­1:27 PM

For the equation the hyperbola, find the coordinates of the center, the foci and the vertices and the equations of the asymptotes. Then graph.11.) (y - 3)2 - (x - 2)2 = 1 16 4

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Apr 5­1:32 PM

For the equation of each parabola, find the coordinates of the vertex and focus, and the equations of the directrix and axis of symmetry. Then graph the equation.12.) x2 + 8x + 4y + 8 = 0

Apr 5­1:36 PM

13.)

14.) Explain a way in which you might distinguish the equationof a parabola from the equation of a hyperbola.

(y - 6)2 = 4x