Geometry (final)

• Parabola


  The set of all points that are equidistant from a fixed point (called the focus) and a fixed straight line (called the directrix)
• Parabola


  • The line through the focus and perpendicular to the directrix is the principal axis of the parabola
  • The point of intersection of the parabola and its principal axis is called the vertex. The vertex is halfway between the focus and the directrix on the principal axis
  • The line segment through the focus connecting two points on the parabola and perpendicular to the principal axis is called the latus rectum. It is also called as the focal width of the parabola
  • The parabola has no center
  • The curve is symmetric about the principal axis
• Parabola opening to the RIGHT


  Equation: y^2 = 4cx

  Focus → (c, 0)

  Directrix → x = -c

  A → (c, 2c)

  B → (c, -2c)
• Parabola opening to the RIGHT


  • Find the equation of the parabola whose focus is at (3,0) and vertex at the origin. Sketch the graph.
  • Find the equation of the parabola whose focus is at (6,0) and vertex at the origin. Sketch the graph.
  • Find the equation of the parabola whose directrix is the line x = -2 and vertex at the origin. Sketch the graph.
• Parabola opening to the LEFT


  Equation: y^2 = -4cx

  Focus → (-c, 0)

  Directrix → x = c

  A → (-c, 2c)

  B → (-c, -2c)
• Parabola opening to the LEFT


  • Find the equation of the parabola whose vertex is at the origin and directrix is the line x = 3. Sketch the graph.
  • Find the equation of the parabola whose focus is at (-4,0) and vertex at the origin. Sketch the graph.
• Parabola opening UPWARDS


  Equation: x^2 = 4cy

  Focus → (0, c)

  Directrix → y = -c

  A → (2c, c)

  B → (-2c, c)
• Parabola opening UPWARDS


  • Find the equation of the parabola whose focus is at (0,3) and vertex at the origin. Sketch the graph.
  • Find the equation of the parabola whose focus is at (0, 5/2) and vertex at the origin. Sketch the graph.
  • Find the equation of the parabola whose directrix is the line y = -2 and vertex at the origin. Sketch the graph.
• Parabola opening DOWNWARDS


  Equation: x^2 = -4cy

  Focus → (0, -c)

  Directrix → y = c

  A → (2c, -c)

  B → (-2c, -c)
• Parabola opening DOWNWARDS


  • Find the equation of the parabola whose focus is at (0,-4) and vertex at the origin. Determine the directrix and sketch the graph.
  • Find the equation of the parabola whose focus is at (0, -5/2) and vertex at the origin. Sketch the graph.
  • Find the equation of the parabola whose directrix is the line y = 3 and vertex at the origin. Sketch the graph.
• ECCLESIASTES 9:10: 'Whatever work you do, do your best. This is because you are going to the grave. There is no working, no planning, no knowledge and no wisdom there.'