Conic Sections

1
Conic Sections
2
Introduction

• Consider a cone being intersected with a plane

Note the different shaped curves that result
3
Introduction
They can be described or defined as a set of
points which satisfy certain condtitions

• We will consider various conic sections and how
  they are described analytically
• Parabolas
• Hyperbolas
• Ellipses
• Circles

4
Parabola

• Definition
• A set of points on the plane that are equidistant
  from
• A fixed line (the directrix) and
• A fixed point (the focus) not on the directrix

focus
directrix
5
Parabola

• Note the line through the focus, perpendicular to
  the directrix
• Axis of symmetry
• Note the point midway between the directrix and
  the focus
• Vertex

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Equation of Parabola

• Let the vertex be at (0, 0)
• Axis of symmetry be y-axis
• Directrix be the line y -p (where p gt 0)
• Focus is then at (0, p)
• For any point (x, y) on the parabola

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Equation of Parabola

• Setting the two distances equal to each other
• What happens if p lt 0?
• What happens if we have

. . . simplifying . . .
Link to web example
8
Working with the Formula

• Given the equation of a parabola
• y ½ x2
• Determine
• The directrix
• The focus
• Given the focus at (-3,0) and the fact that the
  vertex is at the origin
• Determine the equation