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Conics Circle EllipseDefinition The locus of points on a plane that

are a certain distance from a centralpoint.

The locus of points for which the sumof the distances from each point totwo fixed points is equal.

Graph Horizontal:

Vertical:

Rule x+y=r 1

x y

a b+ =

Alternate

form of rule

14 4

x y+ =

is the same as;x+y=4

3x+5y=15

is the same as;

15 3

x y+ =

Parts -center -major axis: line joining 2 vertex-semi-major axis: line joiningcenter to 1 vertex-minor axis: line joining 2co-vertex

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-semi-minor axis: line joiningcenter to 1 co-vertex-foci-center-vertices

-co-vertices-focal radii(L1,L2)

Parameters r=radius a: length from center to vertex on xaxis.b: length from center to vertex ony-axis.

Formulasfor parts

none Horizontal:major axis= 2aminor axis= 2bL1+L2= 2afoci(c)= c=a-b

Vertical:major axis= 2bminor axis= 2aL1+L2= 2bfoci(c)= c=b-a

Properties none Major axis= L1+L2Comparison A circle is an ellipse whos major and minor axis are equal.

Conics Hyperbola ParabolaDefinition The locus of points for which the

difference of the distances from twogiven points is a constant.

The locus of points equidistant from a

fixed line and a fixed point not on theline.

Graph Horizontal: Horizontal, positive:

focus : (h, k+c)directrix : y=k-c

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Vertical: Horizontal, negative :

focus : (h, k-c)

directrix : y=k+c

Vertical, positive :

focus : (h+c, k)directrix : x=h-c

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Vertical, negative :

focus : (h-c, k)

directrix : x=h+cRule Horizontal:

1

x y

a b =

Vertical:

1

x y

a b = or

1

y x

b a =

Horizontal:y=a(x-h)+kpositive; a>0negative; a0negative; a

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axis. h: translation left or right (x-axis)k: translation up or down (y-axis)(h,k) is the vertex.

Formulasfor parts

Horizontal:Transverse axis= 2a

Conjugate axis= 2bVertical:Transverse axis= 2bConjugate axis= 2aGeneral:Foci(c): c=a+bAsymptotes; y=(b/a)x and y=(-b/a)x

Focus;1

4c

a=

Properties Difference between L1 and L2=transverse axis.

L1=L2

Comparison Same formula as an ellipse (whenlooking at horizontal hyperbolas), but

with a negative sign instead of a +.

A parabola looks like a curvierhyperbola without asymptotes.