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Pre-Calc
Conics
www.njctl.org
2015-03-24
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Table of Contents
Review of Midpoint and Distance Formulas
Parabolas
Circles
Ellipses
Hyperbolas
Recognizing Conic Sections from the General Form
Intro to Conic Sections
click on the topic to go to that section
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Midpoint and DistanceFormula
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The Midpoint FormulaGive points A(x1,y1) and B (x2,y2), the point midway between A and B is
Examples: Find the midpoint of the segment with the given endpoints.
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Midpoint and Distance Formula
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1 Find the midpoint of K(1,8) & L(5,2).
A (2,3)
B (3,5)
C (-2,-3)
D (-3,-5)
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Midpoint and Distance Formula
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2 Find the midpoint of H(-4 , 8) & L(6, 10).
A (1,9)
B (2,18)
C (-2,-18)
D (-1,-9)
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Midpoint and Distance Formula
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3 Given the midpoint of a segment is (4 , 9) and one endpoint is (-3 , 10), find the other midpoint.
A (-10 , 8)
B (11 , 8)
C (-10 , 11)
D (.5 , 9.5)
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Midpoint and Distance Formula
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4 What is the distance between (2, 4) and (-1, 8)?
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Midpoint and Distance Formula
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5 What is the distance between (0, 7) and (5, -5)?
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Midpoint and Distance Formula
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Note: The distance between points A and B can be notated as AB
Midpoint and Distance Formula
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6 Given A( 4, 5) and B(x, 1) and AB=5, find all of the possible values of x.
A -7
B -5
C -3
D -1
E 0
F 1
G 3
H 5
I 7
J 9
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Midpoint and Distance Formula
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Intro to Conic Sections
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Conic Sections come from cutting through 2 cones, which is called taking cross sections.
Conic Sections are often times not functions because they do not pass the Vertical Line Test.
Intro to Conic Sections
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A Circle comes from cutting parallel to the "base".
The term base is mis-leading because cones continue on, like lines.
Intro to Conic Sections
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An Ellipse comes from cutting skew (diagonal) to the "base".
Intro to Conic Sections
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A Parabola comes from cutting the cone an intersecting the "base" and parallel to a side.
Intro to Conic Sections
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A Hyperbola comes from cutting the cones perpendicular to the "bases".
This is the only cross section that intersects both cones.
Intro to Conic Sections
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Parabolas
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As we've studied earlier, Parabolas come from a quadratic equation of the form y=ax2+bx+c and have a "U" shaped graph.
Another helpful form of the equation is called Standard Form.Standard Form is
(x - h)2 = 4p(y - k) ,
where (h,k) is the vertex. This is also called Vertex Form.
Example: What is the vertex of: (x - 4)2 = -3(y - 5)
(x + 7)2 = 2(y - 2) (x -3)2 = y
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Parabolas
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7 What is the vertex of
A (3, 2)
B (-3, -2)
C (2, 3)
D (-2, -3)
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Parabolas
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8 What is the vertex of
A (3, 2)
B (-3, -2)
C (2, 3)
D (-2, -3)
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Parabolas
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9 What is the vertex of
A (3, 2)
B (-3, -2)
C (2, -3)
D (-2, -3)
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Parabolas
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10 What is the vertex of
A (3, 2)
B (-3, 2)
C (2, 3)
D (-2, -3)
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Parabolas
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11 What is the vertex of
A (3, 2)
B (-3, -2)
C (2, 3)
D (-2, -3)
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Parabolas
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Converting from General Form to Standard Form
Note: To convert into Standard Form, we use a process called Completing the Square.
Steps:
1) Group the quadratic and its linear term on one side, and move the other linear and constant terms to the other side.
2) If there is a number in front of the quadratic, factor it out of the group.
3) Take the number in front of the linear term, divide it in half and square it.
4) Add this number inside the parenthesis; multiply it by the number you factored out in step two, and add it to the other side of the equation as well.
5) Factor the quadratic function inside the parenthesis
Parabolas
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Parabolas
Example: Find the vertex of the parabola
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17
A (4, 5)
B (-4, 5)
C (-5, 4)
D (5, 4)
What is the vertex of y2 - 10y - x + 29 = 0?
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Parabolas
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18 What is the vertex of
A (4, 5)
B (-4, 5)
C (-5, 4)
D (5, 4)
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Parabolas
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Converting from General Form to Standard Form
}
+18
}-12
Parabolas
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19 What should be factored out of (4y2 - 8y + ___) = x - 9 + ___ ? Te
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Parabolas
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20 What value completes the square of 4(y2 - 2y + ___) = x - 9 + ___ ? Te
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rTe
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r
Parabolas
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21 What value should follow "-9" in 4(y2 - 2y + ___) = x - 9 + ___ ? Te
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rTe
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r
Parabolas
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22 Which is the correct standard form of4(y2 - 2y + ___) = x - 9 + ___
A
B
C
D
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Parabolas
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23 What should be factored out of (-5x2 - 20x + ___) = y - 7 + ___ ? Te
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Parabolas
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24 What value completes the square of -5(x2 + 4x + ___) = y - 7 + ___ ?
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25 What value should follow "-7" in -5(x2 + 4x + ___) = y - 7 + ___?
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Parabolas
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26 Which is the correct standard form of(-5x2 - 20x + ___) = y - 7 + ___
A
B
C
D
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Parabolas
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Focus and Directrix of a Parabola
Axis of Symmetry
Directrix
Focus
Every point on the parabola is the same distance from the directrix and the focus.
L1
L2
L1=L2
The focal distance is the distance from the vertex to the focus, which is the same as the distance from the vertex to the directrix.
Parabolas
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Directrix
Focus
L1
L2
L1=L2
Eccentricity of a Parabola
Parabolas
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Parts of a ParabolaWhether a quadratic has the x2 or y2, they have the same parts.
Directrix
Axis of Symmetry
Vertex
FocusVertex
Focus
Directrix
Axis of Symmetry
cy2+dy+bx+e=0ax2+bx+dy+e=0
Parabolas
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Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
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Parabolas
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Graph the equation from the last example.Parabolas
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Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
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Parabolas
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Graph Teac
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Parabolas
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GraphTe
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Parabolas
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27 Given the following equation, which direction does it open?
A UP
B DOWN
C LEFT
D RIGHT
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Parabolas
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28 Where is the vertex for the following equation?
A (-3 , 4)
B (3 , 4)
C (4 , 3)
D (4 , -3)
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Parabolas
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29 What is the equation of the axis of symmetry for the following equation?
A y = 3
B y = -3
C x = 4
D x = -4
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Parabolas
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30 What is the focal distance in the following equation?
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Parabolas
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31 What is the equation of the directrix for the following equation?
A y = 2
B y = -4
C x = 3
D x = -5
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Parabolas
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32 Where is the focus for the following equation?
A (-3 , 5)
B (3 , 5)
C (5 , 3)
D (5 , -3)
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Parabolas
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33 What is the eccentricity of the following conic section?
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Parabolas
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42 Where is the vertex for the following equation?
A (0 , 4)
B (0 , -4)
C (4 , 0)
D (-4 , 0)
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Parabolas
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43 What is the equation of the axis of symmetry for the following equation?
A y = 0
B y = -0
C x = 4
D x = -4
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Parabolas
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44 What is the focal distance in the following equation?Te
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Parabolas
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45 What is the equation of the directrix for the following equation?
A y = 0
B y = -4
C x = 8
D x = 0
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Parabolas
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46 Where is the focus for the following equation?
A (4 , 8)
B (-4 , 4)
C (4 , 4)
D (4 , -4)
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Parabolas
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47 What is the eccentricity of the following conic section?Te
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Parabolas
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Circles
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48 Write the equation of the circle with center (5 , 2) and radius 6
A
B
C
D
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Circles
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49 Write the equation of the circle with center (-5 , 0) and radius 7
A
B
C
D
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Circles
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50 Write the equation of the circle with center (-2 , 1) and radius
A
B
C
D
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Circles
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51 What is the center and radius of the following equation?
A
B
C
D
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Circles
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53 What is the center and radius of the following equation?
A
B
C
D
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Circles
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54 What is eccentricity of a circle?
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Circles
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Ex: Write the equation of the circle that meets the following criteria:
Diameter with endpoints (4 , 7) and (-2 , -1).
Since the midpoint of the diameter is the center use the midpoint formula.
The radius is distance from the center to either of the given points.
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Circles
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Ex: Write the equation of the circle that meets the following criteria:
Center (1 , -2) and passes through (4 , 6)
Since we know the center we only need to find the radius. The radius is the distance from the center to the point.
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Circles
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Ex: Write the equation of the circle that meets the following criteria:Center at (-5 , 6) and tangent to the y-axis.
"Tangent to the y-axis" means the circle only touches the y-axis at one point. Look at the graph.
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Circles
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Write the equation of the circle in standard form that meets the following criteria:
Complete the square for the x's
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Circles
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55 What is the equation of the circle that has a diameter with endpoints (0 , 0) and (16 , 12)?
A
B
C
D
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Circles
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56 What is the equation of the circle with center (-3 , 5) and contains point (1, 3)?
A
B
C
D
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Circles
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57 What is the equation of the circle with center (7 , -3) and tangent to the x-axis?
A
B
C
D
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Circles
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An ellipse is the set of points the same total distance from 2 points.
In this example,
As point moves along theellipse, L1 and L2 will changebut their sum will stay ten.
P
Ellipses
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P
In this graph F1 and F2 are foci. (Plural of focus)
They lie on the major axis. (The longest distance)
The shortest distance is the minor axis.
Where the axes intersect is the ellipse'scenter.
The more elongated the ellipse the closerthe eccentricity is to 1. The closer an ellipse is to being a circle, the closer theeccentricity is to 0. (0 < e < 1)
Ellipses
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AB
CD E
60 What letter or letters corresponds with ellipse's center?
A
B
C
D
E
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Ellipses
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AB
CD E
61 What letter or letters corresponds with ellipse's foci?
A
B
C
D
E
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Ellipses
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AB
CD E
62 What letter or letters corresponds with ellipse's major axis?
A
B
C
D
E
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Ellipses
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63 Which choice best describes an ellipse's eccentricity?
A e = 0
B 0< e < 1
C e = 1
D e > 1
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Ellipses
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64 What is the center of
A (9 , 4)
B (5 , 6)
C (-5 , -6)
D (3 , 2)
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Ellipses
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65 How long is the major axis of
A 9
B 4
C 3
D 2
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Ellipses
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66 How long is the minor axis of
A 9
B 4
C 3
D 2
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Ellipses
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67 Name one foci of
A
B
C
D
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Ellipses
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68 Name one foci of
A
B
C
D
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Ellipses
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Graphing an Ellipse
· Find and graph the center · Find the length and direction of the major and minor axes· From the center go half the length the axis from the center for each· Graph the ellipse
The center is (4 , -2)The major axis is 6 units and horizontalThe minor axis is 4 units and vertical
Ellipses
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What is equation of an ellipse with foci (3, -2) and (3 , 6) and minor axis of length 8?
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Ellipses
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69 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, what is the center of the ellipse?
A (8 , 2)
B (0 , 2)
C (0 , 1)
D (-8 , 1)
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Ellipses
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70 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, in which direction is the ellipse elongated?
A horizontally
B vertically
C obliquely
D it is not elongated
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Ellipses
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71 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, how far is it from the center to an endpoint of the major axis?
A 10
B 100
C 5
D 25
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Ellipses
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72 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis?
A
B
C
D
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Ellipses
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73 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, find a.
A
B
C
D
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Ellipses
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Converting to Standard Form· complete the square for x and/or y· factor the x's and y's· divide by the constant
Ex:
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Ellipses
Ex:
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75 Convert the following ellipse to standard form.
A
B
C
D
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Ellipses
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The standard form of a horizontal hyperbola is
Hyperbolas
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The standard form of a vertical hyperbola is
Hyperbolas
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To graph a hyperbola in standard form:· graph (h,k) as center of graph
· go a right and left of the center, and b up and down
· make a rectangle through the four points from previous step
· draw asymptotes that contain the diagonals of the rectangle
· decide if hyperbola goes left & right or up & down left & right: the "x term" is first up & down: the "y term" is first
· graph hyperbola
Hyperbolas
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Example: Graph
What are the slopes of the asymptotes? How does this relate to a and b? Why?
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The cente r of the rectangle is?
From the cente r move le ft/right?
From the cente r move up/down?
The hyperbola opens?
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Example: Graph
The hyperbola opens?
From the cente r move up/down?
From the cente r move le ft/right?
The cente r of the rectangle is?
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Hyperbolas
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Standard Form of an Hyperbola
The Foci are equidistant from the center in the horizontal direction if the x-term comes first, or in the vertical direction if the y-term comes first.
The distance from the center to the foci is
In this example, the focal distance is
And their location is at and
Hyperbolas
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85 What is the focal distance for the following equation?
A 12B 13C 5D 8
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Hyperbolas
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86 What is the location of one of the foci for this hyperbola?
A (-13,-6)B (10,-6)C (-10,-6)D (13,-6)
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Hyperbolas
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87 What is the location of one of the foci for this hyperbola?
A (3,7)B (3,19)C (3,-7)D (3,13)
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Hyperbolas
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Recognizing Conic Sections
from theGeneral Form
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