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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.
2
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).
2 2
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
3
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).
4
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)
5
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.
6
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
7
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
8
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
9
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)
10
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.
11
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).
12
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.)
13
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
14
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