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Based on Chapter 11: Analytic Geometry from the 8th edition of Algebra & Trigonometry, Enhanced with Graphing Utilities by Michael Sullivan and Michael Sullivan III. Table of Contents: Section 11.1: cards 1 - 4; Section 11.2: cards 5 - 12; Section 11.3: cards 13 - 24; Section 11.4: cards 25 - 33; Section 11.5: cards 34 - 38; Section 11.6: cards 39 - 46; Section 11.7: cards 47 - 51

1

Definition of circles

When the plane is perpendicular to the axis of the cone when it intersects

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2

Definition of ellipses

When the plane is tilted slightly when it intersects the cone

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3

Definition of parabolas

When the plane is tilter farther so that it is parallel to the cone generator

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4

Definition of hyperbolas

When the plane intersects both parts (upper and lower nappes) of the cone

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5

Parabola

A collection of all points in a plane that are the same distance from a fixed point (focus) as they are from a fixed line (directrix)

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6

Axis of symmetry

Imaginary line that goes through the vertex and perpendicular to the directrix

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7

Vertex of a parabola

*(-{b/2a}, f(-{b/2a}))*

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8

Directrix

A line that is *±2a* units from the vertex

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9

Focal length

The value *a*, the directed distance from the vertex to the focus of the parabola

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10

Latus Rectum

A line segment that passes through the focus and is parallel to the directrix, with the length of *|4a|*

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11

Characteristics of a parabola with the vertex *(h, k)* and a vertical axis of symmetry

Equation:

*(x-h)²=4a(y-k)*If

*a>0*: opens upIf

*a<0*: opens downVertex:

*(h, k)*Focus:

*(h, k±a)*Directrix:

*y=k±a*Axis of symmetry:

*x=h*

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12

Characteristics of a parabola with the vertex *(h, k)* and a horizontal axis of symmetry

Equation:

*(y-k)²=4a(x-h)*If

*a>0*: opens rightIf

*a<0*: opens leftVertex:

*(h, k)*Focus:

*(h±a, k)*Directrix:

*x=h±a*Axis of symmetry:

*y=k*

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13

Ellipse

A collection of all points in the plane whose distance from two fixed points is a constant sum

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14

Center

The point on the focal axis midway between the foci

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15

Foci

Plural of focus, the fixed points equidistance form the center and located on the focal axis

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16

Vertices

The points where the ellipse intersects its axes

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17

Major axis

The line connecting vertices that passes through the center of an ellipse and through its two foci with the length of *2a*

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18

Minor axis

The line between the center of an ellipse that is perpendicular to the major axis

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19

Pythagorean relations

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20

Characteristics of an ellipse with the center *(0, 0)* and the major *x*-axis

Equation:

*{x²/b²} + {y²/b²} = 1*Focal axis:

*x*-axisFoci:

*(±c, 0)*Vertices:

*(±a, 0)*Pythagorean relation:

*a²=b²+c²*

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21

Characteristics of an ellipse with the center *(0, 0)* and the major *y*-axis

Equation:

*{y²/a²} + {x²/b²} = 1*Focal axis:

*y*-axisFoci:

*(0, ±c)*Vertices:

*(0, ±a)*Pythagorean relation:

*a²=b²+c²*

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22

Characteristics of an ellipse with the center *(h, k)* and the major *x*-axis

Equation:

*{(x-h)²/a²} + {(y-k)²/b²} = 1*Focal axis:

*y=k*Foci:

*(h±c, k)*Vertices:

*(h±a, k)*Pythagorean relation:

*a²=b²+c²*

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23

Characteristics of an ellipse with the center *(h, k)* and the major *y*-axis

Equation:

*{(y-k)²/a²} + {(x-h)²/b²} = 1*Focal axis:

*x=h*Foci:

*(h, k±c)*Vertices:

*(h, k±a)*Pythagorean relation:

*a²=b²+c²*

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24

Eccentricity

*e=c/a*

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25

Hyperbola

The collection of points *P=(x, y)* such that the distance of the distances from *P* to the foci is *±2a*

*d(F₁, P) - d(F₂, P) = ±2a*

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26

Foci

Fixed points of the hyperbola

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27

Transverse axis

The line through the foci

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28

Center

The point on the transverse axis midway between foci

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29

Vertices

The points where the hyperbola intersects the transverse axis

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30

Characteristics of a hyperbola with the center *(0, 0)* and transverse axis along *x*-axis

Equation:

*{x²/a²} - {y²/b²} = 1*Focal axis:

*x*-axisFoci:

*(±c, 0)*Vertices:

*(±a, 0)*Pythagorean relation:

*c²=a²+b²*Asymptotes:

*y = ±{b/a}x*

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31

Characteristics of a hyperbola with the center *(0, 0)* and transverse axis along *y*-axis

Equation:

*{y²/a²} - {x²/b²} = 1*Focal axis:

*y*-axisFoci:

*(0, ±c)*Vertices:

*(0, ±a)*Pythagorean relation:

*c²=a²+b²*Asymptotes:

*y = ±{a/b}x*

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32

Characteristics of a hyperbola with the center *(h, k)* and horizontal transverse axis

Equation:

*{(x-h)²/a²} - {(y-k)²/b²} = 1*Focal axis:

*y = h*Foci:

*(h±c, 0)*Vertices:

*(h±a, 0)*Pythagorean relation:

*c²=a²+b²*Asymptotes: (

*y-k) = ±{b/a}(x-h)*

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33

Characteristics of a hyperbola with the center *(h, k)* and vertical transverse axis

Equation:

*{(y-k)²/a²} - {(x-h)²/b²} = 1*Focal axis:

*x = h*Foci:

*(0, k±c)*Vertices:

*(0, k±a)*Pythagorean relation:

*c²=a²+b²*Asymptotes: (

*y-k) = ±{a/b}(x-h)*

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34

General form of a conic

*Ax² + Cy² + Dx + Ey + F = 0; A*and*C*cannot both equal*0*Defines a parabola:

*AC = 0*Defines an ellipse:

*AC*> 0Defines a hyperbola:

*AC < 0*Defines a circle:

*A = C*

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35

General form of a conic with rotation of axes

*Ax² + Bxy + Cy² + Dx + Ey + F = 0*

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36

Coordinates for *(x’, y’)*

*x’ = rcos(α)**y’ = rsin(α)*

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37

The angle to rotate through

*cot(2θ) = {(A-C)/B} *

If *cot(2θ) ≥ 0*: *0° < 2θ ≤ 90°* and *0° < θ < 45°*

If *cot(2θ) < 0*: *90° < 2θ < 180°* and *45° < θ < 90°*

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38

Identifying conics with a rotation of axes theorem

Defines a parabola:

*B² - 4AC = 0*Defines an ellipse (or circle):

*B² - 4AC < 0*Defines a hyperbola:

*B² - 4AC > 0*

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39

Eccentricity

For any point on a conic section, the ration between the distance from the focus and the distance to the directrix

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40

Formula for eccentricity

*e = {d(P₁, F)}/{d(P₂, D)}*

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41

Eccentricity of each conic

Circle:

*e = 0*Parabola:

*e = 1*Ellipse:

*e < 1*Hyperbola:

*e > 1*

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42

Eccentricity of a hyperbola and ellipse

*e = c/a*

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43

Polar equation for a conic where the directrix is perpendicular to the polar axis at a distance of *p* units to the left of the pole

*r = {ep}/{1-ecos(θ)}*

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44

Polar equation for a conic where the directrix is perpendicular to the polar axis at a distance of *p* units to the right of the pole

*r = {ep}/{1+ecos(θ)}*

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45

Polar equation for a conic where the directrix is parallel to the polar axis at a distance of *p* units above the pole

*r = {ep}/{1+esin(θ)}*

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46

Polar equation for a conic where the directrix is parallel to the polar axis at a distance of *p* units below the pole

*r = {ep}/{1-esin(θ)}*

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47

Definition of parametric curves

Let *x = ƒ(t)* and *y = g(t)*, where *ƒ* and *g* are two functions defined on some interval. Then a point on a graph *(x, y)* can be determined by finding *x = ƒ(t)* and *y = g(t)*. Essentially, *(x, y) = (ƒ(t), g(t))*, where *t* is in the domain of *ƒ* and *g*

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48

Orientation of a parametric curve

Successive values of *t* that show movement

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49

Removing the parameter

Choose the easiest equation

Solve for the parameter

Substitute the result for the parameter of the other equation

The resultant equation is the entire graph

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50

Projectile motion

A projectile that is fired at an inclination *θ* to the horizontal, with an initial speed *v₀*, from a height *h* above horizontal

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51

Projectile motion formulas

*x(t) = (v₀cosθ)t**y(t) = -½gt²+(v₀sinθ)t+h**g = 9.8 m/sec²***OR***g = 32 ft/sec²*

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