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Conic Sections
Presented by Greye Dixon
May 7, 2007
What are conic sections?
• Conic sections are lines that define where a flat plane intersects with a double cone, which consists of two cones that meet at one another’s tip.
How can a conic section be drawn?
• A conic section is shown as a graph. It can be shown as a circle, hyperbola, parabola, or ellipse.
• Each section has its own formula for graphing.
• As with all graphing, you use an in-and-out table.
Circles
• Circles are the easiest to figure out and graph out of the four conic sections.
• The formula for the radius of a circle is x2 + y2 = r2, with (0,0) as the center point of the circle.
• The standard form for a circle is r2= ( x - h )2 + ( y - k )2
Parabolas
• The formulas for a parabola with its origin at (0,0) are x2 = 4py for a vertical parabola, and y2 = 4px for a horizontal one.
• It has a directrix, which is a line that is perpendicular to the parabola’s axis on the opposite side of the line that the parabola lies on.
• It also has a fixed point, called a focus, which is the same distance from the vertex of the parabola as the directrix is.
• In a vertical parabola, the focus is at (0,p). In a horizontal parabola, the focus is at (p,0).
Here are graphs of the two types of parabolas:
Hyperbolas
• Hyperbolas are possibly the most difficult part of conic sections, due to both length and complexity.
• The formulas for hyperbolas are • When x comes first, it is a horizontal transverse
axis. • When y comes first, it is a vertical transverse axis.• The vertex is a units from the center, and the foci
are c units from the center. The formula for c is c2 = a2 + b2.
• The formula for asymptotes, the lines that decide the boundaries of the hyperbolas, is x = ± (b/a)y.
Here are the graphs of the two types of hyperbolas:
Ellipses
• The last of the conic sections are ellipses. Ellipses are ovals, so the radii in different points are not completely identical.
• The standard form of a ellipse is , depending on whether it’s horizontal or vertical. When the a comes first, it’s horizontal. When the b comes first, it’s vertical.
• The center can be determined by the formula b2= a2 - c2.
• The foci are either (±c,0), or (0,±c), depending on whether it’s vertical or horizontal.
Here are the graphs of the two types of ellipses:
How are conic sections used in the real world?
• Believe it or not, conic sections really can be used in real-world situations. The circle formula can be used to figure out how long it takes for the blast from a supernova to reach out to certain distances in space.
• The ellipse formula can be used to find out the length and width of a running track.
• The hyperbola formula can be used to figure out the angles of light coming from a lighthouse.
• Parabolas can be used to measure things like suspension bridges.
Peer Assessment
A directrix is a part of which type of conic section? a) ellipses b) parabolas c) hyperbolas d) circles
If the radii are symmetrical, it is a:a) ellipseb) circlec) either
d) none of the above
Peer assessment continued
The only conic section with a minus sign in the formula is a:a) hyperbolab) circlec) parabolad) ellipse
In a parabola the focus is:a) further from the vertex than the directrixb) closer to the vertex than the directrixc) equidistant from the vertex as the directrixd) there is no directrix in a parabola
Sources
All pictures and information came from the following:
• Thinkquest.org• Documents.wolfram.com• www.answers.com