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Conics

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Conics Equation of a circle We should recall that the equation of a circle is given by: Equation of an ellipse What transformation would change the circle into the ... – PowerPoint PPT presentation

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Title: Conics


1
Conics
2
Conic Sections
(1) Circle
A circle is formed when i.e. when the plane ? is
perpendicular to the axis of the cones.
3
Equation of a circle
  • We should recall that the equation of a circle is
    given by

What is the equation of this circle?
4
Conic Sections
(2) Ellipse
An ellipse is formed when i.e. when the plane ?
cuts only one of the cones, but is neither
perpendicular to the axis nor parallel to the a
generator.
5
Equation of an ellipse
  • What transformation would change the circle into
    the ellipse?

What is the equation of the ellipse?
Can you write it another way?
6
Equation of an ellipse
  • Think of an ellipse as a unit circle stretched
    by factor a in the x-direction and factor b in
    the y-direction

This means the equation can be given by
or
Where would it cut the x and y axis?
7
Conic Sections
(3) Parabola
A parabola is formed when i.e. when the plane
? is parallel to a generator.
8
Parabola
  • The general equation of a parabola is given by

9
Conic Sections
(4) Hyperbola
A hyperbola is formed when i.e. when the plane
? cuts both the cones, but does not pass through
the common vertex.
10
Hyperbola (rectangular)
  • What type of graph is the one below? Can you
    write down a general equation for this type of
    graph?

It is a reciprocal graph
The point closest to the origin has co-ordinates
(c,c).
What are the asymptotes of this graph?
11
Hyperbola (rectangular)
  • It can be shown using a matrix transformation
    that after a rotation of 450, our hyperbola has
    the general equation..

What are the asymptotes of this graph?
In a rectangular hyperbola, the asymptotes always
cross each other at right angles.
12
Hyperbola (non-rectangular)
  • The following equation represents a hyperbola
    with centre at the origin and cutting the x-axis
    at a and a.

The asymptotes have equations
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