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Level 504

Ellipses & Circles


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Conic Sections
Curves found by cutting a cone with a plane.
circle cut
if you slice the cone // to the base
ellipse cut
a cut at an angle that doesn't intersect with the base
four conic sections
circle, ellipse, parabola, hyperbola
circle definition
the set of all points on a plane such that each point is at a fixed distance (called a radius) from a fixed point (called the center)
Center
Each regular polygon has a center because it can be inscribed in a circle.
R
radius
standard form of a circle
(x-h)² + (y-k)² = r²
midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
Tangent
tanx=Opp/Adj=y/x
and
When solving an absolute value INEQUALITY, if the original problem has a less than symbol, what type of compound inequality is it?
or
ellipse definition
the set of all points on a plane such that the sum of the distances from the two fixed points (called the foci- singular focus) is constant.
PF1 + PF2
the length of a string. constant. 2a
c
distance from center to focus.
(x-0)²/a² + (y-0)²/b² = 1
standard form of the equation of an ellipse with the center (0,0) with a horizontal major axis
±a
x intercept (of horizontal ellipse with (0,0)
±b
y intercept (of horizontal ellipse with (0,0)
(±a, 0)
vertices (of horizontal ellipse with (0,0)
(0, ±b)
co-vertices (of horizontal ellipse with (0,0)
(±c, 0)
foci (of horizontal ellipse with (0,0)
Eccentricity
c/a
b²=a²-c²
finding a, b or c
what eccentricity is
the distance from c to a
2a
length of major axis
2b
length of minor axis
the foci always lie
on the major axis
the eccentricity must be
less than one and greater than zero
the more circular the ellipse
the closer e is to zero
the flatter the ellipse
the closer e is to one
(h±a,k)
vertices in a horizontal ellipse with center (h,k)
(h,K±b)
covertices in a horizontal ellipse with center (h,k)
(h±c,k)
foci in a horizontal ellipse with center (h,k)
(h,k±a)
vertices in a vertical ellipse with center (h,k)
(h±b,k)
covertices a vertical ellipse with center (h,k)
(h,k±c)
foci a vertical ellipse with center (h,k)
(x-h)²/a² + (y-k)²/b²
Conic Form Horizontal Ellipse
(x-h)²/b² + (y-k)²/a²
Conic Form Horizontal Ellipse
(h±a,k)
Major Vertex Horizontal
(h,k±a)
Major Vertex Vertical
(h,K±b)
Minor Vertex Horizontal
(h±b,k)
Minor Vertex Vertical
(h±c,k)
Horizontal Foci
(h,k±c)
Vertical Foci
(x-h)²/a² - (y-k)²/b²
Conic Form Horizontal Hyperbolas
(y-k)²/a² - (x-h)²/b²
Conic Form Vertical Hyperbolas
y=±b/a
Horizontal Asymptote
y=±a/b
Vertical Asymptote
(h±a,k)
Horizontal Vertices Hyperbola
(h,k±a)
Vertical Vertices Hyperbola