MTH065Elementary Algebra II

Chapter 13Conic Sections

IntroductionParabolas (13.1)

Circles (13.1)Ellipses (13.2)

Hyperbolas (13.3)Summary

Where we’ve been …

• MTH 060 – Linear Functions & Equations• Single Variable: ax + b = 0• Solution: A single real number.

• Two Variables: ax + by = c y = mx + b f(x) = mx + b• Solutions: Many ordered pairs of real numbers.• Graph: A line.

2 2y x

Where we’ve been …

• MTH 065 – Quadratic Functions & Equations• Single Variable: ax2 + bx + c = 0• Solutions: 0, 1, or 2 real numbers

• Two Variables: y = ax2 + bx + c f(x) = ax2 + bx + c f(x) = a(x – h)2 + k• Solutions : Many ordered pairs of real numbers.• Graph: A parabola.

2 512 2y x x

What’s missing …

• Quadratic Equations that may also include a y2 term (not all functions).

Ax2 + By2 + Cx + Dy + E = 0

A, B, C, D, & E are constantsA and B not both 0

Note: Quadratic equations may also include an xy term, but the study of such equations requires trigonometry.

Parabolas

y = ax2 + bx + c• Graphing (complete the square): y = a(x - h)2 + k• Vertex: (h, k)• h = -b/(2a)

• Orientation:• Open upward: a > 0• Open downward: a < 0

• Width:• Narrow: |a| > 1• Wide: |a| < 1

• Graphing: Vertex & One Other Point

2 512 2y x x

212 ( 1) 3y x

Parabolas

x = ay2 + by + c• Graphing (complete the square): x = a(y - k)2 + h• Vertex: (h, k)• k = -b/(2a)

• Orientation:• Open right: a > 0• Open left: a < 0

• Width:• Narrow: |a| > 1• Wide: |a| < 1

• Graphing: Vertex & One Other Point

22 12 19x y y 22( 3) 1x y

Parabolas – Special PropertiesFocus• The point 1/(4a) units from the vertex along the

axis of symmetry and inside the parabola.• Reflective property:• Light or any other wave emitted from the focus will be

reflected in a beam parallel to the axis of symmetry.• A satellite dish, for example, uses this property in

reverse.

1

4p

a

Ellipses

Ax2 + By2 + Cx + Dy + E = 0where A & B are both positive or both negative.

• Graphing form: Complete the squares & set equal to 1

• Center: (h,k)• 4 Vertices: (h ± a, k), (h, k ± b)

2 2

2 2

( ) ( )1

x h y k

a b

Ellipses – Special PropertiesFoci• The two points c units from the center along the

major axis where c2 = a2 – b2 if a > b or c2 = b2 – a2 if a < b.

• Reflective property:• Sound or any other wave emitted from one focus will

be reflected to the other focus.

• Satellites have elliptical orbits with the object being orbited at one of the foci.

Circles – Special Ellipses

• A circle is just an ellipse with a = b and a single “focus” at the center (since c2 = a2 – b2 = 0).

Ax2 + Ay2 + Cx + Dy + E = 0

(x – h)2 + (y – k)2 = r2

• Center: (h, k)• Radius: r

HyperbolasAx2 + By2 + Cx + Dy + E = 0

where A & B have opposite signs.• Graphing form: Complete the squares & set equal to 1

• Center: (h,k)• 2 Vertices: • 1st form: (h ± a, k)• 2nd form: (h, k ± b)

• Asymptotes:

2 2

2 2

( ) ( )1

x h y k

a b

2 2

2 2

( ) ( )1

x h y k

a b

or

( )bay x h k

ba(h,k)

ba(h,k)

Hyperbolas – Special PropertiesFoci• The two points c units from the center inside each

branch, where c2 = a2 + b2 • Reflective property:• Light or any other wave emitted from one focus towards

the other branch will be reflected directly away from the other focus (or vice versa).

• Hyperbolic mirrors are used in reflector telescopes.• Lampshades cast hyperbolic shadows on a wall.

Parabola

Hyperbola

Conic Sections – SummaryAx2 + By2 + Cx + Dy + E = 0

• A ≠ 0 & B = 0• Up/Down Parabola

• A = 0 & B ≠ 0• Left/Right Parabola

• A & B w/ same sign• Ellipse• A = B gives a circle

• A & B w/ opposite signs• Hyperbola

To graph … complete the squares.

More Applications of Conics

• Parabolas• http://www.doe.virginia.gov/Div/Winchester/

jhhs/math/lessons/calc2004/appparab.html

• Ellipses• http://www.doe.virginia.gov/Div/Winchester/

jhhs/math/lessons/calc2004/appellip.html

• Hyperbolas• http://www.doe.virginia.gov/Div/Winchester/

jhhs/math/lessons/calc2004/apphyper.html