Mathematics 116Bittinger

•Chapter 7

•Conics

Mathematics 116

• Conics• A conic is the intersection

of a plane an a double-napped cone.

Degenerate Conic

• Degenerate conic – plane passes through the vertex

• Point

• Line

• Two intersecting lines

Algebraic Definition of Conic

2 2 0Ax Bxy Cy Dx Ey F

Definition of Conic

• Locus (collection) of points satisfying a certain geometric property.

Circle

• A circle is the set of all points (x,y) that are equidistant from a fixed point (h,k)

• The fixed point is the center.

• The fixed distance is the radius

Algebraic def of Circle

• Center is (h,k)

• Radius is r

2 2 2x h y k r

Equation of circlewith center at origin

2 2 2x y r

Def: Parabola

• A parabola is the set of all points (x,y) that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line.

Standard Equation of ParabolaVertex at Origin

• Vertex at (0,0)

• Directrix y = -p

• Vertical axis of symmetry

2 4x py

Standard Equation of ParabolaOpening left and right

• Vertex: (0,0O

• Directrix: x = -p

• Axis of symmetry is horizontal

2 4y px

Willa Cather – U.S. novelist (1873-1947)

•“The higher processes are all simplification.”

Definition: Ellipse

• An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is a constant.

Standard Equation of EllipseCenter at Origin

• Major or focal axis is horizontal

2 2

2 21

x y

a b

Standard Equation of EllipseCenter at Origin

• Focal axis is vertical

2 2

2 21

x y

b a

Ellipse: Finding a or b or c

2 2 2a b c 2 2 2c a b

Definition: hyperbola

• A hyperbola is the set of all points (x,y) in a plane, the difference whose distances from two distinct fixed points (foci) is a positive constant.

Hyperbola equationopening left and right

centered at origin

2 2

2 21

x y

a b

Standard Equation of Hyperbolaopening up and down

centered at origin

2 2

2 21

y x

a b

Hyperbolafinding a or b or c

2 2 2b c a

2 2 2a b c

Objective – Conics centered at origin

• Recognize, graph and write equations of

• Circle

• Parabola

• Ellipse

• Hyperbola– Find focal points

Rose Hoffman – elementary schoolteacher

• “Discipline is the keynote to learning. Discipline has been the great factor in my life.”

Mathematics 116

•Translations

•Of

•Conics

Circle

• Center at (h,k) radius = r

2 2 2x h y k r

Ellipse major axis horizontal

2 2

2 21

x h y k

a b

Ellipsemajor axis vertical

2 2

2 21

x h y k

b a

Hyperbolaopening left and right

2 2

2 21

x h y k

a b

Hyperbolaopening up and down

2 2

2 21

y k x h

a b

Parabolavertex (h,k) opening up and

down

24x h p y k

Parabola vertex (h,k)opening left and right

24y k p x h

Objective

• Recognize equations of conics that have been shifted vertically and/or horizontally in the plane.

Objective

• Find the standard form of a conic – circle, parabola, ellipse, or hyperbola given general algebraic equation.

Example

• Determine standard form – sketch

• Find domain, range, focal points

2 24 6 8 9 0x y x y

Example - problem

• Determine standard form – sketch

• Find domain, range, focal points

2 24 2 16 11 0x y x y

1,2center

Winston Churchill

•“It’s not enough that we do our best; sometimes we have to do what’s required.”