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