10.2 Parabolas

Objective• To determine the relationship

between the equation of a parabola and its focus, directrix, vertex, and axis of symmetry.

• To graph a parabola

Definition• A set of points equidistant from

a fixed point (focus) and a fixed line (directrix).

• 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.

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

Vertical

2( ) 4 ( ), 0x h p y k p

• General Form

If p > 0 opens up, if p < 0 opens down

• 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: 2 4x py

Horizontal parabola

General Form2( ) 4 ( ), 0y k p x h p

If p > 0 opens right, if p < 0 opens left

• Vertex: (h, k)

• Focus: (h + p, k)

• Directrix: x = h-p

• Axis of symmetry: y = k

Example 1

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

Example2Finding the Focus of a Parabola

• Find the focus of the parabola given by

21 12 2

y x x

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)

Example 4

• opens: p = • vertex focus • directrix axis of symmetry

21 ( 2) 34

y x

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.

• 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.

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.

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)

2y x