Introduction to Conic Sections Conic sections will be defined in two different ways in this unit. 1.The set of points formed by the intersection of a plane and a double-napped cone. 2.The set of points satisfying certain conditions in relationship to a fixed point and a fixed line or to two fixed points.

Conic sections are the shapes formed on a plane when that plane intersects two cones (vertex to vertex). We will discuss four different conic sections: circles, parabolas, ellipses, and hyperbolas. These four conic sections can degenerate into degenerate conic sections. The intersections of the double-napped cone and the plane are a point, a line, and intersecting lines.

Section 10.2 Parabolas

1 st Definition of a Parabola A parabola is a conic section formed when a plane intersects one of the cones and is parallel to a diagonal side (generator) of the cone. The degenerate conic section associated with a parabola is a line.

2 nd Definition of a Parabola A parabola is a set of points in a plane that are the same distance from a given point, called the focus and a given line called directrix.

Draw a line through the focus perpendicular to the directrix. This line is the axis of the parabola. Find the point on the axis that is equidistant from the focus and the directrix of the parabola. This is the vertex of what will become a parabola. We call the distance from the focus to the vertex the focal length.

directrix focus vertex focal length axis

In general, the graph of a parabola is bowl- shaped. The focus is within the bowl. The directrix is outside the bowl and perpendicular to the axis of the parabola.

F directrix axis

General Equation of a Parabola Vertical Axis Ax 2 + Dx + Ey + F = 0 Horizontal Axis Cy 2 + Dx + Ey + F = 0 To rewrite from the general form to other forms you will complete the square.

Standard Equation of a Parabola If p = the focal length, then the standard form of the equation of a parabola with vertex at (h, k) is as follows: Vertical Axis (x – h) 2 = 4p(y – k) Horizontal Axis (y – k) 2 = 4p(x – h) 4p = focal width: the length of the perpendicular segment through the focus whose endpoints are on the parabola.

F directrix axis Focal width

Vertex Equation of a Parabola If p = the focal length and (h, k) is the vertex of a parabola, then the vertex form of the equation of a parabola is Vertical Axis y = a(x – h) 2 + k Horizontal Axis x = a(y – k) 2 + h where

Example 1 For each parabola state the form of the given equation, horizontal or vertical, find the vertex, axis, the focal length, focus, directrix, and focal width. Graph the ones indicated.

1.4(x − 2) = (y + 3) 2 Graph. form: vertex: axis: focal length: focus: directrix: focal width: (2, −3) Standard and horizontal

V F 4p = 4 so p = 1 4(x − 2) = (y + 3) 2

1.4(x − 2) = (y + 3) 2 Graph. form: vertex: axis: focal length: focus: directrix: focal width: (2, −3) Standard and horizontal y = −3 4p = 4, p = 1 (3, −3) x = 1 4p = 4

V F

2.2x 2 + 4x – y − 3 = 0 form: y + 3 = 2x 2 + 4x y __ = 2(x 2 + 2x + __ ) y = 2(x 2 + 2x + 1) y + 5 = 2(x + 1) 2 y = 2(x + 1) 2 − 5 (vertex form) General and vertical

vertex: axis: focus: directrix: (−1, −5) x = −1 y = 2(x + 1) 2 − 5

3.x 2 + 2y − 6x + 8 = 0 Graph. Form: 2y + 8 = −x 2 + 6x 2y + 8 − 9 = −(x 2 − 6x + 9) 2y − 1 = −(x − 3) 2 General and Vertical

vertex: axis: focal length:focal width: focus: directrix: x = 3 4p = 2

V F Graph.

Example 2 Write the equation for each parabola.

1.Vertex (2, 4); Focus (2, 6) in standard form p = 2 vertical parabola 4p = 8 (x – 2) 2 = 8(y – 4)

2.Focus (−2, 0); Directrix: x = 4 in vertex form 2p = 6 so p = 3 horizontal parabola Vertex: (−2 + 3, 0) = (1, 0)

3.Vertex (4, 3); Parabola passes through (5, 2) and has a vertical axis. Write in standard form. (x – 4) 2 = 4p(y – 3) (5 – 4) 2 = 4p(2 – 3) 1 = −4p −1 = 4p (x − 4) 2 = −(y – 3)