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Level 498

Conic Sections II


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Ellipse
Equation adds x²/a² and y²/b²=1
Focus
A fixed points
Vertices of an Ellipse
The endpoints on the major axis of an ellipse.
Co-vertices of an Ellipse
The endpoints of the minor axis of an ellipse.
Foci of Ellipse
c^2 = a^2 - b^2
Standard Form Equation of an Ellipse
(x-h)^2/a^2 + (y-k)^2/b^2 = 1; horizontal; a>b>o.
Hyperbola
Equation subtracts x²/a and y²/b=1; x and y coefficients have different signs.
Transverse Axis
The axis that goes through the vertices of a hyperbola.
Standard Form Equation of an Hyperbola
(x-h)^2/a^2 - (y-h)^2/b^2 = 1; horizontal; Asymptote: y=+/-bx/a
Foci of Hyperbola
c^2 = a^2 + b^2
Parabola
the graph of a quadratic function
|a| = 1/4c
Focus/Directrix of a Parabola
Standard Form Equation of a Parabola
y = a(x-h)^2 + k; Vertical; vertex (h,k)
Standard form (Circle)
(x-h)^2 + (y-k)^2 = r^2
(h, k)
Center of a circle
Distance Formula
(Square root)(x2-x1)^2+(y2-y1)^2
Standard form (Parabola)
y = (1/4c)(x-h)^2+k
(Y): (h,k)
Vertex of a parabola
x=h
Axis Of symmetry
Y: (h,k+c)
Focus of a Parabola
Directrix
y=k-c
Laterus Rectum
Ab. Value 4C
Standard form (Ellipse)
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
(h,k)
Center of a ellipse
2a
Major Axis (Ellipse)
2b
Minor Axis(ellipse)
vertices (ellipse)
Endpoints of the major axis
Co-Verticies (ellipse)
Endpoints of the minor axis
Foci (Ellipse)
Foci are on the major axis 'c' units from the center
LR (ellipse)
Width of ellipse at foci
2b^2/a
LR (Hyperbola)
2a
Transversal axis (hyperbola)
2b
Conjugate axis (hyperbola)
Vertices (hyperbola )
endpoints of the transversal axis
Foci (hyperbola)
c^2 = A^2 + b^2
Circle
r= asin(theta)
The squared terms have equal coefficients
Defining an equation to produce a circle
(x-h)²+(y-k)²=r²
The standard form for a circle
Center
Each regular polygon has a center because it can be inscribed in a circle.
radius
From center to a point on the circle.
Defining an equation to produce an ellipse
The squared terms have different coefficients with the same sign
(x-h)²/a²+(y-k)²/b²=1
Standard equation of ellipse
c²=|a²-b²|
Equation for finding the foci of an ellipse
Defining an equation to produce a hyperbola
The squared terms have coefficients with different signs
(x-h)²/a²-(y-k)²/b²=1
standard equation of a hyperbola
-(x-h)²/a²+(y-k)²/b²=1
Standard form form a hyperbola when the x-term is negative
The center
(h,k) in the standard forms for a hyperbola represents...
c²=a²+b²
The equation for finding the foci of a hyperbola
(y-k)=±(b/a)(x-h)
The equation for finding the asymptotes of a hyperbola
The asymptote of a hyperbola represents
lines that restrict the width of a given hyperbolic equation
The transverse axis of a parabola
The sum of the distances from the focus to points on the parabola perpendicular to the axis of symmetry from the focus. It always equals 4p
Defining an equation to produce a parabola
One term is squared and the other is not
4p(y-k)=(x-h)²
The standard form for a parabola when the x-term is squared
4p(x-h)=(y-k)²
The standard form for a parabola when the y-term is squared
the vertex
(h,k) in the standard forms for a parabola represents...
|p|
The distance from the vertex from the focus and the directrix
The directrix of a parabola
A line perpendicular to the axis of symmetry and a distance of |p| away from the vertex
the parabola faces upwards
Under the conditions p+ and x²
the parabola faces downwards
Under the conditions p- and x²
Under the conditions p+ and y²
the parabola faces to the right
Under the conditions p- and y²
The parabola faces to the left
As p gets smaller...
the parabola will tighten