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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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introduction to conic sections
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.
  • The set of points satisfying certain conditions in relationship to a fixed point and a fixed line or to two fixed points.
slide2

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.
section 10 2

Section 10.2

Parabolas

1 st definition of a parabola
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.
2 nd definition of a parabola
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.
slide8

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.

slide9

focal length

focus

axis

vertex

focal length

directrix

slide10

In 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.

slide11

F

axis

directrix

general equation of a parabola
General Equation of a Parabola
  • 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 parabola
Standard Equation of a 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.

slide14

Focal width

F

axis

directrix

vertex equation of a parabola
Vertex Equation of a Parabola
  • 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
Example 1
  • 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.
slide17

4(x− 2) = (y + 3)2 Graph.

  • form:
  • vertex:
  • axis:
  • focal length:
  • focus:
  • directrix:
  • focal width:

Standard and horizontal

(2, −3)

slide19

4(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

slide20

F

V

slide21

2. 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

slide22

vertex:

  • axis:
  • focus:
  • directrix:

(−1, −5)

x = −1

slide24

3. x2 + 2y − 6x + 8 = 0 Graph

  • Form:
  • 2y + 8 = −x2 + 6x
  • 2y + 8 − 9 = −(x2 − 6x + 9)
  • 2y − 1 = −(x − 3)2

general

slide25

vertex:

  • axis:
  • focal length: focal width:
  • focus:
  • directrix:

x = 3

4p = 2

example 2
Example 2
  • Write the equation for each parabola.
slide27

1. Vertex (2, 4); Focus (2, 6) in standard form

  • p = 2
  • vertical parabola
  • (x – 2)2 = 8(y – 4)
slide28

2. Focus (−2, 0); Directrix: x = 4 in vertex form

  • 2p = 6 so p = 3
  • horizontal parabola
  • Vertex: (−2 + 3, 0) = (1, 0)
slide29

3. 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)
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