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.”