math chapter 10 - conics

how to know if circle (Ax²+Bxy+Cy²+Dx+Ey+F=0)

A = C

how to know if ellipse (Ax²+Bxy+Cy²+Dx+Ey+F=0)

A and C have the same sign but not equal to each other

what does the Bxy term do (Ax²+Bxy+Cy²+Dx+Ey+F=0)

rotation of axes occurs

circle definition

a set of coplanar points equidistant from a fixed point (center)

standard form circle

(x-h)²+(y-k)²=r²

center: (h,k)

radius = r

how to graph circle

get into standard form

center point and four points (N,S,E,W)

ellipses definition

the set of coplanar points such that the sum of the distances from any point P on the ellipse to two fixed points (the foci) is a constant (2a)

major axis

2a

minor axis

2b

the end points of major axis are

vertices

the end points of minor axis are

co vertices

ellipse: a equals

length of center to vertex or half of major axis

ellipse: b equals

length of center to co vertex or half minor axis

ellipse: c equals

length center to focus

ellipse: a,b,c equation

a²-b²=c²

eccentricity means and equals

helps to determine shape of ellipse

e = c/a

always 0<e<1

as e—>0 ellipse approaches the shape of a circle

as e—>1 ellipse become mor elongated

how to know which denominator a and b go in for ellipse

a on first one: ellipse is horizontal and the x axis is the major axis

• so b is the second one

b on the first one: ellipse is vertical and the y axis is the major axis

• so a is the second one

length of latus rectum equation

l.r = 2b²/a

each latus rectum is a segment of the above length, the major axis, with a focus as its midpoint

how to graph ellipse

exact coordinates for

• center

• vertices

• co vertices

• foci

• endpoints of each latus rectum

• eccentricity states

how to know if hyperbola (Ax²+Bxy+Cy²+Dx+Ey+F=0)

if A and C have opposite signs

hyperbola definition

a set of coplanar points such that the absolute value of the difference in the distances from a point P to two fixed points (foci) is a constant (2a)

transverse axis

2a

conjugate axis

2b

endpoints of transverse axis

vertices

for hyperbola the circle is intersection of what

transverse and conjugate axis

hyperbola: a equals

length of semi transverse axis

center to vertez

hyperbola: b equals

length of semi conjugate axis

center to either endpoint of conjugate axis

hyperbola: c equals

center to focus

hyperbola: a,b,c equation

a² + b² = c²

standard form of hyperbola

(x-h)²/a² - (y-k)²/b² = 1

(y-k)²/a² - (x-h)²/b² = 1

how to know denominator a and b for hyperbola

it always stays the same a first then b

when transverse axis is x axis

opens left and right

when transverse axis is y axis

opens up and down

horizontal transversal axis then asymptote is

y-k = +-b/a (x-h)

vertical transversal axis then asymptote is

y-k = +-a/b(x-h)

how to graph hyperbola

• center (h,k)

• vertices (endpoints of transverse axis)

• endpoints of conjugate axis

• foci

• eccentricity

• equations for two asymptotes

conjugate hyperbola definitions

conjugate axis of one hyperbola is the transverse axis of another

equilateral hyperbola definition

occurs when a=b and the asymptotes are perpendicular to each other

parabola definition

a set of coplanar points equidistant from a fixed point (focus) and a fixed line (directrix)

how to know if parabola (Ax²+Bxy+Cy²+Dx+Ey+F=0)

if A or C = 0

equations for parabola

horizontal: (y-h)²=4p(x-k)

vertical: (x-h)²=4p(y-k)

parabola vertex point is what

(h,k)

parabola a equals

vertex to focus

vertex to directrix

parabola 2a equals

focus to endpoint of latus rectum

parabola 4a equals

length of latus rectum

parabola directrix is what and equation

it is a line not a point on the graph

x= ________

y=________

parabola axis of symmetry definition

line through focus and perpendicular to directrix