Notes 8.1 Conics Sections – The Parabola
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Transcript of Notes 8.1 Conics Sections – The Parabola
Notes 8.1 Notes 8.1
Conics Sections –Conics Sections – The Parabola The Parabola
I. IntroductionI. Introduction
A.) A.) A conic section is the intersection of A conic section is the intersection of
a plane and a conea plane and a cone..B.) B.) By changing the angle and the By changing the angle and the
location of intersection, a parabola, location of intersection, a parabola, ellipse, hyperbola, circle, point, line, or ellipse, hyperbola, circle, point, line, or a pair of intersecting lines is a pair of intersecting lines is produced.produced.
C.) Standard Conics:C.) Standard Conics:
1.) Parabola1.) Parabola
2.) Ellipse2.) Ellipse
3.) Hyperbola3.) Hyperbola
D.) Degenerate ConicsD.) Degenerate Conics
1.) Circle1.) Circle
2.) Point2.) Point
3.) Line3.) Line
4.) Intersecting Lines4.) Intersecting Lines
E.) Forming a Parabola – E.) Forming a Parabola – When a plane intersects a double-napped When a plane intersects a double-napped cone and is parallel to the side of the cone, cone and is parallel to the side of the cone, a parabola is formed.a parabola is formed.
2 2 0Ax Bxy Cy Dx Ey F
If both B and C = 0, or A and B = 0, the conic is a parabola
F.) General Form Equation for All Conics
II. The ParabolaII. The Parabola
2
2
or
y ax bx c
x ay by c
A.) In general - A parabola is the graph of a quadratic equation, or any equation in the form of
B.) Def. - A PARABOLA is the set of all points in a plane equidistant from a particular line (the DIRECTRIX) and a particular point (the FOCUS) in the plane.
Axis of Symmetry
Focus Focal Width
Focal Length
Directrix
Vertex
axisx
x p ,0p 0, p
axisy
2 4y px
p
y p
2 4x pyStandard Form
Focus
Directrix
Axis of Symmetry
Focal Length
Focal Width 4 p
p
4 p
C.) Parabolas (Vertex = (0,0))
D.) Ex. 1- Find the focus, directrix, and focal width of the parabola y = 2x2.
1
8y
10,
8
1
8p
14
2p
2 1
2x y Focus =
Directrix =
Focal Width = 1
2
E.) Ex. 2- Do the same for the parabola
1x
4
1,0
1p
4 4 p
2 4y xFocus =
Directrix =
Focal Width =
21
4x y
F.) Ex. 3- Find the equation of a parabola with a directrix of x = -3 and a focus of (3, 0).
212x y
24 3 x y
24 px y
y k
x h p ,h p k ,h k p
x h
24y k p x h
p
y k p
24x h p y k St. Fm.
Focus
Directrix
Ax. of Sym.
Fo. Lgth.
Fo. Wth. 4 p
p
4 p
G.) Parabolas (Vertex = (h, k))
H.) Ex. 4- Find the standard form equation for the parabola with a vertex of (4, 7) and a focus of (4, 3).
3 7 4p
4x
24 4 7x p y
24x h p y k
ax. of sym.
24 16 7x y
I.) Ex. 5- Find the vertex, focus, and directrix of the parabola 0 = x2 – 2x – 3y – 5.
1, 2
51,
4
21 3 2x y
2 2 1 3 5 1x x y
2 2 3 5x x y
11
4y
vertex =
Directrix =
focus =
III. Paraboloids of III. Paraboloids of RevolutionRevolution
A.) A PARABOLOID is a 3-dimensional solids created by revolving a parabola about an axis.
Examples of these include headlights, flashlights, microphones, and satellites.
B.) Ex. 6– A searchlight is in the shape of a B.) Ex. 6– A searchlight is in the shape of a paraboloid of revolution. If the light is 2 paraboloid of revolution. If the light is 2 feet across and 1 ½ feet deep, where feet across and 1 ½ feet deep, where should the bulb be placed to maximize should the bulb be placed to maximize the amount of light emitted?the amount of light emitted?
2 4x py
The bulb should be placed 2” from the vertex of the paraboloid
1
6p
2 31 4
2p 0,0
31,
2
31,
2
0, p