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Introduction to Conic Sections - PowerPoint PPT Presentation

Introduction to Conic Sections. Conic sections will be defined in two different ways in this unit. The set of points formed by the intersection of a plane and a double-napped cone.

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Presentation Transcript

• Conic sections will be defined in two different ways in this unit.

• The set of points formed by the intersection of a plane and a double-napped cone.

• The set of points satisfying certain conditions in relationship to a fixed point and a fixed line or to two fixed points.

• Conic sections are the shapes formed on a plane when that plane intersects two cones (vertex to vertex). We will discuss four different conic sections: circles, parabolas, ellipses, and hyperbolas.

• These four conic sections can degenerate into degenerate conic sections. The intersections of the double-napped cone and the plane are a point, a line, and intersecting lines.

http://youtu.be/GDHNoQHQmtQ

Section 10.2

Parabolas

1stDefinition of a Parabola

• A parabola is a conic section formed when a plane intersects one of the cones and is parallel to a diagonal side (generator) of the cone.

• The degenerate conic section associated with a parabola is a line.

2nd Definition of a Parabola

• A parabola is a set of points in a plane that are the same distance from a given point, called the focus and a given line called directrix.

• Draw a line through the focus perpendicular to the directrix. This line is the axis of the parabola. Find the point on the axis that is equidistant from the focus and the directrix of the parabola. This is the vertex of what will become a parabola. We call the distance from the focus to the vertex the focal length.

focal length directrix. This line is

focus

axis

vertex

focal length

directrix

• In directrix. This line is general, the graph of a parabola is bowl-shaped. The focus is within the bowl. The directrix is outside the bowl and perpendicular to the axis of the parabola.

F directrix. This line is

axis

directrix

General Equation of a Parabola directrix. This line is

• Vertical Axis

• Ax2 + Dx + Ey + F = 0

• Horizontal Axis

• Cy2 + Dx + Ey + F = 0

• To rewrite from the general form to other forms you will complete the square.

Standard Equation of a directrix. This line is Parabola

If p = the focal length, then the standard form of the equation of a parabola with vertex at (h, k) is as follows:

Vertical Axis

(x – h)2 = 4p(y – k)

Horizontal Axis

(y – k)2 = 4p(x – h)

4p = focal width: the length of the segment through the focus whose endpoints are on the parabola.

Focal width directrix. This line is

F

axis

directrix

Vertex Equation of a Parabola directrix. This line is

• If p = the focal length and (h, k) is the vertex of a parabola, then the vertex form of the equation of a parabola is

• Vertical Axis

• y = a(x – h)2 + k

• Horizontal Axis

• x= a(y – k)2 + h

• where

Example 1 directrix. This line is

• For each parabola state the form of the given equation, horizontal or vertical, find the vertex, axis, the focal length, focus, directrix, and focal width. Graph the ones indicated.

• 4( directrix. This line is x− 2) = (y + 3)2 Graph.

• form:

• vertex:

• axis:

• focal length:

• focus:

• directrix:

• focal width:

Standard and horizontal

(2, −3)

4 directrix. This line is p = 4 so p = 1

F

V

• 4( directrix. This line is x− 2) = (y + 3)2 Graph.

• form:

• vertex:

• axis:

• focal length:

• focus:

• directrix:

• focal width:

Standard and horizontal

(2, −3)

y = −3

4p = 4, p = 1

(3, −3)

x = 1

4p = 4

F directrix. This line is

V

• 2. directrix. This line is 2x2+ 4x – y− 3 = 0

• form:

• y + 3 = 2x2 + 4x

• y + 3 + __ = 2(x2 + 2x + __ )

• y + 3+ 2 = 2(x2 + 2x + 1)

• y + 5 = 2(x + 1)2

• y = 2(x + 1)2 − 5 (vertex form)

General and vertical

• vertex: directrix. This line is

• axis:

• focus:

• directrix:

(−1, −5)

x = −1

• 3. directrix. This line is x2 + 2y − 6x + 8 = 0 Graph

• Form:

• 2y + 8 = −x2 + 6x

• 2y + 8 − 9 = −(x2 − 6x + 9)

• 2y − 1 = −(x − 3)2

general

• v directrix. This line is ertex:

• axis:

• focal length: focal width:

• focus:

• directrix:

x = 3

4p = 2

Example 2 directrix. This line is

• Write the equation for each parabola.

• 1. directrix. This line is Vertex (2, 4); Focus (2, 6) in standard form

• p = 2

• vertical parabola

• (x – 2)2 = 8(y – 4)

• 2. directrix. This line is Focus (−2, 0); Directrix: x = 4 in vertex form

• 2p = 6 so p = 3

• horizontal parabola

• Vertex: (−2 + 3, 0) = (1, 0)

• 3. directrix. This line is Vertex (4, 3); Parabola passes through (5, 2) and has a vertical axis. Write in standard form.

• (x – 4)2 = 4p(y – 3)

• (5 – 4)2 = 4p(2 – 3)

• 1 = -4p

• -1 = 4p

• (x − 4)2 = -(y – 3)