math ch15: conic sections

represents all points in a plane that are equidistant from a fixed point called the focus and a specific line called the directrix

the distance from the vertex to the focus is usually denoted by

the distance from the vertex to the focus is equal to the —— and is called the

the chord of the parabola which passes through the focus is called the — or —-

the focal chord is denoted by

standard form of equations for parabolas: opens vertically

standard form of equations for parabolas: opens horizontally

when a graph opens up: p

when a graoh opens down: p

when a graph opens left: p

when a graph opens right: p

vertex of vertical equations:

vertex of horizontal equations:

focus of vertical equations:

focus of horizontal equations:

axis of symmetry of vertical equations:

axis of symmetry of horizontal equations:

directrix of vertical equations:

directrix of horizontal equations:

focal chord of vertical equations

focal length of horizontal equations

focal length of vertical equations

focal length of horizontal equations

all points in a plane such that the sum of the distance from two points, called foci, is constant

symbol from vertex to center

symbol from covertex to center

symbol from focus to center

symbol from vertex to vertex

symbol from covertex to covertex

symbol from focus to focus

standard form of equation for horizontal ellipse

standard form of equations for vertical ellipse

orientation of horizontal ellipse

orientation of vertical ellipse

center of horizontal ellipse

center of vertical elipse

foci of horizontal ellipse

foci of vertical ellipse

vertices of horizontal ellipse

verticies of vertical ellipse

covertices of horizontal ellipse

covertices of vertical ellipse

major axis of horizontal ellipse

major axis of vertical elipse

a b c relationship:

eccentricity for horizontal ellipse

eccentricity for vertical ellipse

represents the distance between one of the foci and the center of the ellipse

as the foci move closer together c and e values approach

when the eccentricity reaches 0, the ellipse is a ——— and a and b are equal to the

equation of the circle with a center (hk) and radius r is

the equation x³=y²=r² is when the center is

(x1-h)² + (y1-k)² = r², then (x1,y1) is

(x2-h)²+(y2-k)²<r, then (x2,y2) is

(x3-h)²+(y3-k)²>r², then (x3,y3) is

all points in a plane such that absolute value of the difference of the distances from the two foci is constant results in a

a hyperbola has —- asxes of symmetry

this axis has a length of 2a units and connects to the vertices

this axis is perpendicular to the transverse, passes through the center, and has a length of 2b units

symbol from vertex to center

symbol from vertex to conjugate point

it is used to draw

symbol from focus to center

distance from vertex to vertex

it is the full length of the

distance from conjugate point to conjugate point

full length of the —- and helps to form a

distance from focus to focus symbol

total distance between

standard form of hyperbola equations from horizontals:

standard form of hyperbola equations from verticals

orientation of horizontal hyperbola

orientation of vertical hyperbola

center of horizontal hyperbola

center of vertical hyperbola

vertices of horizontal hyperbola

vertices of vertical hyeprbola

foci of horizontal hyperbola

foci of vertical hyperbola

transverse axis of horizontal hyperbola

transverse axis of vertical hyperbola

conjugate axis of horizontal hyperbola

conjugate axis of vertical hyperbola

asymptotes of horizontal hyperbola

asymptotes of vertical hyperbola

a b and c relationships of hyperbola

eccentricity of hyperbola

you can determine the type of conic by using the equation in general from by how many ways

method 1: discriminant method

if the discriminant is less than 0, b=0 and a=c, the conic is a

if the discriminant is less than 0 and either b is not equal to 0 and a is not equal to c, the conic is an

if the discriminant is equal to 0, the conic is a

if the discriminant is greater than 0, the conic is a

method two is described by the