Algebra Class: Parabolas and Their Properties

Definition & Standard Equations

• A parabola is the locus of points where the distance from a fixed point (focus) is equal to the distance from a fixed line (directrix).
• The line perpendicular to the directrix passing through the focus is called the axis of symmetry.
• The intersection point of the parabola and its axis is called the vertex.
• A chord perpendicular to the axis is called a double ordinate. The double ordinate passing through the focus is called the latus rectum.

Standard Parabolas

• For parabolas with vertex at (0, 0) and a > 0:
  • y² = 4ax: opens to the right, focus (a, 0), directrix x = -a, length of latus rectum is 4a
  • y² = -4ax: opens to the left, focus (-a, 0), directrix x = a, length of latus rectum is 4a
  • x² = 4ay: opens upwards, focus (0, a), directrix y = -a, length of latus rectum is 4a
  • x² = -4ay: opens downwards, focus (0, -a), directrix y = a, length of latus rectum is 4a

Example 1: y² = -12x

• Focus: (-3, 0)
• Vertex: (0, 0)
• Directrix: x = 3
• Axis of symmetry: y = 0
• Length of latus rectum: 12

Example 2: x² = 6y

• Focus: (0, 3/2)
• Vertex: (0, 0)
• Directrix: y = - 3/2
• Axis of symmetry: x = 0
• Length of latus rectum: 6

Example 3: y² = 4ax with double ordinate of length 8a

• The ends of the double ordinate are (x₁, 4a) and (x₁, -4a)
• Since these points lie on the parabola, they satisfy the equation y² = 4ax.
• Using this, we find x₁ = 4a.
• The slope of the line connecting the vertex to one end of the double ordinate is 4a/4a = 1.
• The slope of the line connecting the vertex to the other end is -4a/4a = -1.
• The product of these slopes is -1, which means the lines are perpendicular.