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B) Calculus > Conics > Flashcards

Flashcards in Conics Deck (49):
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Parabolas

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

1

Parabola Focus

The point from which the parabola curve originates

2

Directrix (d)

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

3

Ellipses

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

4

Foci

The two points that determine the shape of the ellipse

5

Major axis

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

6

Minor axis

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

7

Ellipse Center

Always (0,0)

8

Hyperbolas

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

9

Vertices

Hyperbola points on the x-axis

10

Asymptotes

Hyperbola points on the y-axis

11

x-value for Horizontal Ellipses

x=√([1-(y^2/b^2)]*a^2)

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

12

y-value of horizontal ellipses

y=√([1-(x^2/a^2)]*b^2)

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

13

Foci of horizontal ellipses (without 'b'

c=√(a^2-[y^2/(1-x^2/a^2)])

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

14

Vertices of horizontal ellipses (without given foci)

a=√(x^2/[1-y^2/b^2])

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

15

Foci of vertical ellipses (without 'b'

c=√(a^2-[x^2/(1-y^2/a^2)])

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

16

y-value of vertical ellipses

y=√([1-(x^2/b^2)]*a^2)

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

17

x-value for vertical Ellipses

x=√([1-(y^2/a^2)]*b^2)

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

18

Slanted asymptotes for horizontal hyperbolas

y=±bx/a

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

19

Horizontal Axis Hyperbolas

Hyperbola curves in towards x=0

20

Vertical axis hyperbolas

Hyperbola curves in towards y=0

21

Slanted asymptotes for vertical asymptotes

y=±ax/b

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

22

Foci of horizontal hyperbolas (without 'b'

c=√(a^2+(y^2)/[(x^2)/(a^2)-1])

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

23

Foci of vertical hyperbolas (without 'b'

c=√(a^2+(x^2)/[(y^2)/(a^2)-1])

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

24

Vertices of vertical ellipses (without given foci)

a=√(y^2/[1-x^2/b^2])

a- vertices at (0,±a)
b- given value

25

Vertices of vertical hyperbolas (without given foci)

a=√(y^2/[1+x^2/b^2])

a- vertices at (0,±a)
b- given value

26

Vertices of horizontal hyperbolas (without given foci)

a=√(x^2/[1+y^2/b^2])

a- vertices at (±a,0)
b- given value

27

X-value of horizontal hyperbolas

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

B-value for hyperbolas

b= √(c^2-a^2)

29

B-value for ellipses

b= √(a^2-c^2)

30

Y-value for vertical hyperbolas

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

Y-value of horizontal hyperbolas

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

General x,y,c,a formulas

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

PL-value

Distance from a given point on the parabola to the directrix
Use, [d-r*cosθ]

34

PF-value (r)

Distance from a given point on the parabola to the focus point
Basically a 'radius'
Use r=ε*d/(1+ε*cosθ)

35

Eccentricity (ε)

Ratio of the PF-value to the PL-value

ε=|PF|/|PL|

36

Identifying Conic Sections from eccentricity

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

Conic sections

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

38

Circle eccentricity

ε=0

39

Polar equation for d>(x=0)

r=ε*d/(1+ε*cosθ)

40

Polar equation for d<(x=0)

r=ε*d/(1-ε*cosθ)

41

Polar equation for d>(y=0)

r=ε*d/(1+ε*sinθ)

42

Polar equation for d<(y=0)

r=ε*d/(1-ε*sinθ)

43

Ellipsoid formula

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

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

44

Elliptic paraboloid formula

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

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

45

One-sheet hyperboloid formula

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

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

46

Two-sheet hyperboloid

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

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

47

Elliptic Cone formula

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

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

48

Hyperbolic paraboloid formula

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

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