Conic Sections

The Parabola

Introduction

• Consider a ___________ being intersected with a __________

Introduction

• We will consider various conic sections and how they are described analytically

– Parabolas– Hyperbolas– Ellipses– Circles

Parabola

• Definition– A set of points on the plane that are

equidistant from – A fixed line

(the ____________) and – A fixed point

(the __________) not on the directrix

Parabola

• Note the line through the focus, perpendicular to the directrix– Axis of symmetry

• Note the point midway between the directrix and the focus– ______________

Equation of Parabola

• Let the vertex be at (0, 0)– Axis of symmetry be y-axis– Directrix be the line y = -p (where p > 0)– Focus is then at (0, p)

• For any point (x, y) on the parabola

( , )x y

Equation of Parabola

• Setting the two distances equal to each other

• What happens if p < 0?• What happens if we have

2 20x y p y p . . . simplifying . . .

2 4 ?y p x

Working with the Formula

• Given the equation of a parabola– y = ½ x2

• Determine– The directrix– The focus

• Given the focus at (-3,0) and the fact that the vertex is at the origin

• Determine the equation

When the Vertex Is (h, k)

• Standard form of equation for vertical axis of symmetry

• Consider– What are the coordinates

of the focus?– What is the equation

of the directrix?

2______ 4 __________x p

(h, k)

When the Vertex Is (h, k)

• Standard form of equation for horizontal axis of symmetry

• Consider– What are the coordinates

of the focus?– What is the equation

of the directrix?

2 ___________y k x h

(h, k)

Try It Out

• Given the equations below, – What is the focus?– What is the directrix?

2( 3) ( 2)x y

2 4 9 0x y y

24 12 12 7 0x x y

Another Concept

• Given the directrix at x = -1 and focus at (3,2)

• Determine the standard form of the parabola

Assignment

• See Handout• Part A 1 – 33 odd• Part B 35 – 43 all

Conic Sections

The EllipsePart A

Ellipse

• Another conicsection formedby a plane intersecting acone

• Ellipse formed when

Definition of Ellipse

• Set of all points in the plane …– ___________ of distances from two fixed

points (foci) is a positive _____________

Definition of Ellipse

• Definition demonstrated by using two tacks and a length of string to draw an ellipse

Graph of an EllipseNote various parts

of an ellipse

Deriving the Formula

• Note– Why?

• Write withdist. formula

• Simplify

( , )P x y

1 2( , ) ( , ) 2d P F d P F a

Major Axis on y-Axis

• Standard form of equation becomes

• In both cases– Length of major axis = _______– Length of __________ axis = 2b–

2 2

2 2 1x yb aa b

2 2 2c a b

Using the Equation

• Given an ellipse with equation

• Determine foci• Determine values for

a, b, and c• Sketch the graph

2 2

136 49x y

Find the Equation

• Given that an ellipse …– Has its center at (0,0)– Has a minor axis of length 6– Has foci at (0,4) and (0,-4)

• What is the equation?

Ellipses with Center at (h,k)

• When major axis parallelto x-axis equation can be shown to be

Ellipses with Center at (h,k)

• When major axis parallelto y-axis equation can be shown to be

Find Vertices, Foci

• Given the following equations, find the vertices and foci of these ellipses centered at (h, k)

2 2( 6) ( 2) 125 81x y

2 29 6 36 36 0x y x y

Find the Equation

• Consider an ellipse with– Center at (0,3)– Minor axis of length 4– Focci at (0,0) and (0,6)

• What is the equation?

Assignment

• Ellipses A• 1 – 43 Odd