Parabolas conics 7.1

Conic sections

figures formed when a plane intersects a double napped cone

Degenerate Conic

Formed when a plane intersects a double-napped cone at their vertices

Basic conic sections (4)

Parabolas, Circles, Ellipses, Hyperbolas

Types of degenerate conics

Point, Line, 2 intersecting lines

Parabola

All points P are equidistant from a fixed point, F, as they are from a fixed line, D.

Directrix

A line outside the parabola that all points are equidistant to the focus

Standard Form for a Parabola (vertical)

(x-h)^2\=4p(y-k)

Standard Form for a Parabola (horizontal)

(y-k)^2\=4p(x-h)

if p\>0 (for a vertical parabola)

opens up

if p

opens down

if p\>0 (for a horizontal parabola)

opens right

if p

opens left

focus (equation)

(h,k+p) or (h+p,k)

Latus Rectum of Parabola

the line segment on the parabola parallel to the directrix which has endpoints 2𝑝 units away from the focus

what is another name for the latus rectum

focal diameter of a parabola

Parabola (h,k) form (vertical)

(y-k)\=a(x-h)^2

Parabola (h,k) form (horizontal)

(x-h)\=a(y-k)^2

what does "a" equal? (h,k form)

a\=1/4p

what does "a" equal? (standard form)

a\=4p

Ellipse

All points p whose distances from the two fixed points called the foci is a constant

Which axis contains the foci

The major axis

What is the distance from the foci to the center

C

Verticies

The points of intersection of the major axis

Co-Verticies

The points of intersections of the minor axis

A,b,c relationship (ellipse)

C^2 \= a^2-b^2

eccentricity

Determines how "circular"or "stretched" an ellipse will be

What values will eccentricity always be between?

0 and 1

Eccentricity is the ratio of...?

C to a

What does the value c represent? (Eccentricity)

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

Eccentricity e\=...

E\=c/a

For any circle, wha is the eccentricity?

0

standard form of the equation of a circle

(x-h)^2 + (y-k)^2 \= r^2

Hyperbola

the collection of all points of whose distances from two fixed points, called the foci, is a constant

transverse axis (hyperbola)

a line that goes through the center and passes through the foci, vertices, and center

foci (hyperbola)

(h+-c, k) or (h, k+-c)

conjugate axis (hyperbola)

the line perpendicular to the transverse axis and through the center

vertices (hyperbola)

are a distance "a" from the center

asymptotes (horizontal hyperbola)

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

asymptotes (vertical hyperbola)

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

a,b,c relationship (hyperbola)

c^2 \= a^2 + b^2

what is the eccentricity for a hyperbola?

always \>1

if B^2-4ac

circle

if b^2-4ac

ellipse

if b^2-4ac\=0

parabola

if b^2-4ac\>0

hyperbola

what is LORAN?

Long Range Navigation