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Parabola - Coggle Diagram
Parabola
Intro
Conic section
formed by intersecting double right circular cone with a plane
Conic section as a locus of point
Point moves in plane: distance from focus to perpendicular from directrix is constant.
fixed point -> focus fixed line -> directrix
eccentricity
is a constant ratio
types
Ellipse
e < 1
Hyperbola
e > 1
Parabola
e = 1
Equation of conic
is a general second degree equation
Terms
Axis of conic
passing through focus perpendicular to directrix
Vertex
focal chord
any chord passing through focus
Double ordinate
st line perepndicular to axis
latus rectum
combo of double oridnate and focal chord
length = 2
e
(dist of S from directrix)
Condition for parabola
h^2 - ab = 0
abc + 2fgh + .... =/= 0
Geometry
Point
Position of a point wrt para
Inside -> concave
y^2 - 4x < 0
Outside - > convex
y^2 - 4x > 0
Line
General
Intersection of line
Line touches parabola
D = 4(4a^2 - 4 amc)
no meeting
line intersects at 2 points
Segment geometry
Chord
Equation of chord of parabola bisected at given point
Slope of line joining P(t1) and Q(t2)
2/t1 + t2
t1*t2
always constant
t1*t2 = - 4 when P and Q subtend right angle at vertex
t1*t2 = -1 when it passes through focus
Focal chord
Focus splits focal chord in 2 segments
Properties
Focal chord length: a(t + 1/t)^2 for P(t)
Focal chord segments' harmonic mean is semi-latus rectum.
Parabola focal chord P(t1)Q(t2) implies t1*t2 = -1
Tangent
Normal
Forms of equations
Standard
Vertex form
vertex not at origin
axis parallel to x axis
axis parallel to y axis
Simple form
Vertex at origin
Types
x^2 = 4ay
x^2 = - 4ay
y^2 = 4ax
y^2 = - 4ax
Using focus and directrix
h^2 - ab = 0
Parametric
properties