10.2 Parabolas

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10.2 Parabolas
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Objective

• To determine the relationship between the
  equation of a parabola and its focus, directrix,
  vertex, and axis of symmetry.
• To graph a parabola

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Definition

• A set of points equidistant from a fixed point
  (focus) and a fixed line (directrix).

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• The midpoint between the focus and the directrix
  is called the vertex.
• The line passing through the focus and the vertex
  is called the axis of the parabola.
• A parabola is symmetric with respect to its axis.

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• p is the distance from the vertex to the focus
  and from the vertex to the directrix.

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Vertical
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• General Form

If p gt 0 opens up, if p lt 0 opens down
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• Vertex (h, k)
• Focus (h, k p)
• Directrix y k p
• Axis of symmetry x h
• If the vertex is at the origin (0, 0), the
  equation is

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Horizontal parabola
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General Form
If p gt 0 opens right, if p lt 0 opens left
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• Vertex (h, k)
• Focus (h p, k)
• Directrix x h-p
• Axis of symmetry y k

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Example 1

• Find the standard equation of the parabola with
  vertex (3, 2) and focus (1, 2)

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Example2Finding the Focus of a Parabola

• Find the focus of the parabola given by

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Example 3Finding the Standard Equation of a
Parabola

• Find the standard form of the equation of the
  parabola with vertex (1, 3) and focus (1, 5)

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Example 4

• opens p
• vertex focus
• directrix axis of symmetry

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Application

• A line segment that passes through the focus of a
  parabola and has endpoints on the parabola is
  called a focal chord. The focal chord
  perpendicular to the axis of the parabola is
  called the latus retum.

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• A line is tangent to a parabola at a point on the
  parabola if the line intersects, but does not
  cross, the parabola at the point.
• Tangent lines to parabolas have special
  properties related to the use of parabolas in
  constructing reflective surfaces.

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Reflective Property of a Parabola

• The Tangent line to a parabola at a point P makes
  equal angles with the following two line
• The line passing through P and the focus
• The axis of the parabola.

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Example 5Finding the Tangent Line at a point on
a Parabola

• Find the equation of the tangent line to the
  parabola given by
• At the point (1, 1)

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