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B) Calculus > Conics > Flashcards

Conics Flashcards

0
Q

Parabola Focus

A

The point from which the parabola curve originates

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1
Q

Parabolas

A

Set of points equidistantly spaced from a fixed point and a fixed line

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2
Q

Directrix (d)

A

The line from which all points on the parabola are equally spaced
y-value for foci on y-axis
X-value for foci on x-axis

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3
Q

Ellipses

A

A closed curve on which all points are spaced from two points the sum of which are always equal to the same value

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4
Q

Foci

A

The two points that determine the shape of the ellipse

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5
Q

Major axis

A

The greater distance from an individual focus for a point on an elipse

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6
Q

Minor axis

A

The lesser distance from an individual focus for a point on an elipse

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7
Q

Ellipse Center

A

Always (0,0)

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8
Q

Hyperbolas

A

Set of points on whose distances from two fixed points have a constant difference
Look like two parabolas approaching each other
Form slanted asymptotes

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9
Q

Vertices

A

Hyperbola points on the x-axis

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10
Q

Asymptotes

A

Hyperbola points on the y-axis

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11
Q

x-value for Horizontal Ellipses

A

x=√([1-(y^2/b^2)]*a^2)

a- vertices at (±a,0)
b- √(a^2-c^2)
c- foci at (±c,0)

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12
Q

y-value of horizontal ellipses

A

y=√([1-(x^2/a^2)]*b^2)

a- vertices at (±a,0)
b- √(a^2-c^2)
c- foci at (±c,0)

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13
Q

Foci of horizontal ellipses (without ‘b’

A

c=√(a^2-[y^2/(1-x^2/a^2)])

a- vertices at (±a,0)
c- foci at (±c,0)

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14
Q

Vertices of horizontal ellipses (without given foci)

A

a=√(x^2/[1-y^2/b^2])

a- vertices at (±a,0) or (0,±a)
b- given value

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15
Q

Foci of vertical ellipses (without ‘b’

A

c=√(a^2-[x^2/(1-y^2/a^2)])

a- vertices at (0,±a)
c- foci at (0,±c)

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16
Q

y-value of vertical ellipses

A

y=√([1-(x^2/b^2)]*a^2)

a- vertices at (0,±a)
b- √(a^2-c^2)
c- foci at (0,±c)

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17
Q

x-value for vertical Ellipses

A

x=√([1-(y^2/a^2)]*b^2)

a- vertices at (0,±a)
b- √(a^2-c^2)
c- foci at (0,±c)

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18
Q

Slanted asymptotes for horizontal hyperbolas

A

y=±bx/a

b- √(a^2-c^2)
a- vertices at (±a,0)
c- foci at (±c,0)

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19
Q

Horizontal Axis Hyperbolas

A

Hyperbola curves in towards x=0

20
Q

Vertical axis hyperbolas

A

Hyperbola curves in towards y=0

21
Q

Slanted asymptotes for vertical asymptotes

A

y=±ax/b

b- √(a^2-c^2)
a- vertices at (0,±a)
c- foci at (0,±c)

22
Q

Foci of horizontal hyperbolas (without ‘b’

A

c=√(a^2+(y^2)/[(x^2)/(a^2)-1])

a- vertices at (±a,0)
c- foci at (±c,0)

23
Q

Foci of vertical hyperbolas (without ‘b’

A

c=√(a^2+(x^2)/[(y^2)/(a^2)-1])

a- vertices at (0,±a)
c- foci at (0,±c)

24
Q

Vertices of vertical ellipses (without given foci)

A

a=√(y^2/[1-x^2/b^2])

a- vertices at (0,±a)
b- given value

25
Q

Vertices of vertical hyperbolas (without given foci)

A

a=√(y^2/[1+x^2/b^2])

a- vertices at (0,±a)
b- given value

26
Q

Vertices of horizontal hyperbolas (without given foci)

A

a=√(x^2/[1+y^2/b^2])

a- vertices at (±a,0)
b- given value

27
Q

X-value of horizontal hyperbolas

A

x=√([a^2+a^2(y^2/b^2)])

a- vertices at (±a,0)
b- √(c^2-a^2)
c- foci at (±c,0)

28
Q

B-value for hyperbolas

A

b= √(c^2-a^2)

29
Q

B-value for ellipses

A

b= √(a^2-c^2)

30
Q

Y-value for vertical hyperbolas

A

y=√([a^2+a^2(x^2/b^2)])

a- vertices at (0,±a)
b- √(c^2-a^2)
c- foci at (0,±c)

31
Q

Y-value of horizontal hyperbolas

A

Y=√(b^2(x^2/a^2)-b^2)

a- vertices at (±a,0)
b- √(a^2-c^2)
c- foci at (±c,0)

32
Q

General x,y,c,a formulas

A

1=(x^2/a^2)-(y^2/b^2)
Ellipses add the two, with ‘b’ as √(a^2-c^2)
Hyperbolas subtract the two, with ‘b’ as √(c^2-a^2)
Horizontal means x/a and y/b
Vertical means y/a and x/b

33
Q

PL-value

A

Distance from a given point on the parabola to the directrix

Use, [d-r*cosθ]

34
Q

PF-value (r)

A

Distance from a given point on the parabola to the focus point
Basically a ‘radius’
Use r=εd/(1+εcosθ)

35
Q

Eccentricity (ε)

A

Ratio of the PF-value to the PL-value

ε=|PF|/|PL|

36
Q

Identifying Conic Sections from eccentricity

A

If ε=1, the curve is nothing more than a parabola
If ε1, the curve is a hyperbola
If ε=0, the curve is a circle

37
Q

Conic sections

A

Describe curves on a plane running through two, hour-glass stacked cones

38
Q

Circle eccentricity

A

ε=0

39
Q

Polar equation for d>(x=0)

A

r=εd/(1+εcosθ)

40
Q

Polar equation for d<(x=0)

A

r=εd/(1-εcosθ)

41
Q

Polar equation for d>(y=0)

A

r=εd/(1+εsinθ)

42
Q

Polar equation for d<(y=0)

A

r=εd/(1-εsinθ)

43
Q

Ellipsoid formula

A

(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

44
Q

Elliptic paraboloid formula

A

(x^2)/(a^2)+(y^2)/(b^2)=z

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

45
Q

One-sheet hyperboloid formula

A

(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

46
Q

Two-sheet hyperboloid

A

-(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

47
Q

Elliptic Cone formula

A

(x^2)/(a^2)+(y^2)/(b^2)=(z^2)/(c^2)

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

48
Q

Hyperbolic paraboloid formula

A

z=(x^2)/(a^2)-(y^2)/(b^2)

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

B) Calculus (17 decks)

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