A nice interactive demonstration can be found here: http://demonstrations.wolfram.com/ConicSectionsTheDoubleCone/

image came from: http://commons.wikimedia.org/wiki/File:Conic_sections_3.png

Parabola

• definition: the set of all points in a plane whose distances from a fixed point (called the focus) and a fixed line (called the directrix) are equal. The focus and the directrix are in the same plane as the points.
• apply definition to derive the formula, where p is the distance at the vertex
• discuss the physical application of the focus
• example problem 1: Give the standard equation of a parabola whose focus is at: (-6, 2) and whose directrix is at y = 8. Draw a rough sketch and label at least one point other than the vertex with its exact coordinates.
• example problem 2: Let's put the following in standard form and identify the directrix, focus, vertex, and make a rough sketch on your activity page:  5x² - 10x = 4y + 3

Circle

• definition: the set of all points in a plane whose distance from a fixed point is a constant. The fixed point is in the same plane.
• example problem 3: Find the equation described by the set of all points that are a distance of 3 from (-2, 7)

Ellipse

• definition: the set of all points in a plane where the sum of the distances from two fixed points (called foci) is a constant.
• for simplicity, let's start off deriving the formula with the center at the origin, the verticies at ±a, endpoints of minor axis at ±b, and the foci at ±c
  Let's derive the equation.
• Now what if the center is had (h, k)?

Hyperbola

• definition: the set of all points in a plane where the difference of the distances from two foci is a constant.
• Let's find a physical meaning for the difference constant.
• Let's discuss the image to help you remember the significance of the sum and difference constants.

A website with loads of interesting in depth information can be found here: http://www2.andrews.edu/~calkins/math/webtexts/numb19.htm

Optional Information: Eccentricity

The eccentricity can be thought of as a measure of the deviation from circular. In that regard, the eccentricity of

• a circle is zero,
• an ellipse is between zero and one,
• a parabola is one, and
• a hyperbola is greater than one.

You can view a demos of eccentricity at: http://demonstrations.wolfram.com/ConicSection/ or by using my new program at: http://www.lsquaredmath.us/eccentricity.

Another way to think of eccentricity is for any point, P, on the curve of a conic, the distance from P to a focal point divided by the distance from P to a line called the directrix is a constant.  You can view a demo of this at http://demonstrations.wolfram.com/ConicSectionsPolarEquations/.

The book defines eccentricity as c/a where c is the distance from the center to a focal point and a is the distance from the center to a vertex.  I will leave it to you to show that this definition matches the definition above.  You may want to make up some simple examples where the center of the conic is at the origin.  Where would the directrix be in an ellipse or in a hyperbola when our definition above does not include a directrix?

Drawing the Conics

• This webite shows how to draw a parabola and you can alter the method a bit to also get the hyperbola and the ellipse.
• An ellipse is pretty easy to figure out how to draw when you use the definition based on the foci.
• Here is a hyperbola using the two foci definition.