Conics Flashcards
Parabola Focus
The point from which the parabola curve originates
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
Ellipses
A closed curve on which all points are spaced from two points the sum of which are always equal to the same value
Foci
The two points that determine the shape of the ellipse
Major axis
The greater distance from an individual focus for a point on an elipse
Minor axis
The lesser distance from an individual focus for a point on an elipse
Ellipse Center
Always (0,0)
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
Vertices
Hyperbola points on the x-axis
Asymptotes
Hyperbola points on the y-axis
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)
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)
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)
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
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)
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)
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)
Slanted asymptotes for horizontal hyperbolas
y=±bx/a
b- √(a^2-c^2)
a- vertices at (±a,0)
c- foci at (±c,0)
Horizontal Axis Hyperbolas
Hyperbola curves in towards x=0
Vertical axis hyperbolas
Hyperbola curves in towards y=0
Slanted asymptotes for vertical asymptotes
y=±ax/b
b- √(a^2-c^2)
a- vertices at (0,±a)
c- foci at (0,±c)
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)
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)
Vertices of vertical ellipses (without given foci)
a=√(y^2/[1-x^2/b^2])
a- vertices at (0,±a)
b- given value
Vertices of vertical hyperbolas (without given foci)
a=√(y^2/[1+x^2/b^2])
a- vertices at (0,±a)
b- given value
Vertices of horizontal hyperbolas (without given foci)
a=√(x^2/[1+y^2/b^2])
a- vertices at (±a,0)
b- given value
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)
B-value for hyperbolas
b= √(c^2-a^2)
B-value for ellipses
b= √(a^2-c^2)
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)
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)
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
PL-value
Distance from a given point on the parabola to the directrix
Use, [d-r*cosθ]
PF-value (r)
Distance from a given point on the parabola to the focus point
Basically a ‘radius’
Use r=εd/(1+εcosθ)
Eccentricity (ε)
Ratio of the PF-value to the PL-value
ε=|PF|/|PL|
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
Conic sections
Describe curves on a plane running through two, hour-glass stacked cones
Circle eccentricity
ε=0
Polar equation for d>(x=0)
r=εd/(1+εcosθ)
Polar equation for d
r=εd/(1-εcosθ)
Polar equation for d>(y=0)
r=εd/(1+εsinθ)
Polar equation for d
r=εd/(1-εsinθ)
Ellipsoid formula
(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Elliptic paraboloid formula
(x^2)/(a^2)+(y^2)/(b^2)=z
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
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
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
Elliptic Cone formula
(x^2)/(a^2)+(y^2)/(b^2)=(z^2)/(c^2)
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Hyperbolic paraboloid formula
z=(x^2)/(a^2)-(y^2)/(b^2)
a- ±x-vertices
b- ±y-vertices
c- ±z-vertices
Parabolas
Set of points equidistantly spaced from a fixed point and a fixed line
