conic math

conic sections

figures created by specifying a relationship between a set of pts; another point (the focus); and a line (the directrix)

identify parabola by definition

set of pts equidistant from focus and directrix

identify ellipse by definition

set of pts that are half as far from focus as from directrix

identify hyperbola by definition

set of pts 10 times as far from focus as from directrix

identify parabola by equation (B=0)

either A=0 or C=0, but not both

identify circle by equation (B=0)

A=C

identify ellipse by equation (B=0)

A /= C (and neither is 0; same signs)

identify hyperbola by equation (B=0)

either A < 0 or C < 0 but not both (opposite signs)

identify none by equation (B=0)

(degenerate conic) no constant left after completing square

General equation of conic section (B=0)

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identify parabola by equation (B/=0)

If quadratic has one solution, the original conic was a parabola (it represents the axis of symmetry)

identify hyperbola by equation (B/=0)

if quadratic has two solutions, original conic was a hyperbola (they represent the asymptotes)

identify ellipse by equation (B/=0)

if quadratic has no solutions, original conic was a circle or ellipse (it is the center)

general equation after zoomed out

Ax^2 + Bxy + Cy^2=0

general equation after zoomed out in terms of y/x

A+B(y/x)+C(y/x)^2=0 (divide by x^2)

identify circle by discriminant

if B^2 - 4AC<0

* A and C have same sign

identify parabola by discriminant

if B^2 - 4AC = 0

* A or C = 0

identify hyperbola by discriminant

if B^2-4AC>0

* A or C is negatives and neither = 0

Discriminant of conic section

B^2-4AC

eccentricity

* distance from P to focus / distance from P to directrix
* for ellipse or hyperbola: c/a (center to focus/center to vertex)

identify parabola by eccentricity

e=1

identify circle by eccentricity

o<e<1

identify hyperbola by eccentricity

e=0

identify parabola by eccentricity

e>1

parabola

a curve on which all pts are equidistant from focus and the directrix

circle

a set of all pts in a plane that are a distance r (radius) from a given point (center)

ellipse

set of all pts in a plane, the sum of whose distances from 2 distinct fixed pts (foci) is constant

* D1+D2 is a constant

hyperbola

the difference of the distances between foci and a pt is constant

* D2-D1 is a positive constant

standard form of circle

standard form of ellipse

standard form of hyperbola

asymptotes of hyberbola

standard form of parabola (horizontal)

if y^2, then parabola opens right or left

standard form of parabola (vertical)

if x^2, then parabola opens up or down

two focus property of Ellipse

an ellipse is a collection of points P such that d(P,F1) + d(P,F2)=2a (major axis)

two focus property of Hyperbola

set of all points P so d(P,F1) - d(P,F2)=2a (transverse axis)

Why must a parabola hv an eccentricity of 1?

definition of parabola: pts are equidistant from the focus and directrix

eccentricity formula

why must e of ellipse be less than 1? hyperbola > 1? (e=c/a)

in an ellipse, c<a

* creates a fraction of c/a

why does e tell us about shape of ellipse

how flat or round it is

* if flat, closer to 1
* if round, closer to 0
* in a circle, c=0 (a^2-a^2= 0)

degenerate form of hyperbola

asymptotes

degenerate form of ellipse

degenerate form of circle

x^2+y^2=r^2 -→ x^2+y^2=0

* circle with radius 0

degenerate form of parabola

if B=0

* A=0 and C=0

while in a regular parabola

* if B=0, then A or B=0 not both

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left with

Dx+Ey+F=0

* linear equation

degenerate conics are created when

zoom out of graph of conic section (let x and y approach infinity)

conic discriminant

looking at what’s left of graph when x and y approach infinity

degenerate hyperbola

asymptotes of hyperbola (can also find asymptotes thru discriminant)

degenerate circle or ellipse

its center

degenerate parabola

axis of symmetry