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

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:

Horizontal parabola

General Form 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)

Example2 Finding the Focus of a Parabola Find the focus of the parabola given by

Example 3 Finding 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

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 5 Finding 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)