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