The planets of the solar system orbit about the sun.
The orbits of the planets are nearly circles, but are actually ellipses.
Circles and ellipses are both conic sections. The other conic sections are parabolas and hyperbolas.
Conic sections are so named because they can be formed by slicing a cone with a plane in various ways.
The type of conic section formed depends on the angle of the plane relative to the cone.
A coordinate system can be superimposed on the plane so that each conic section is represented by an equation in standard form.
Each of the standard forms can be transformed into the general form, so that one type of equation represents all of the conic sections.
With this tool, you can investigate the connection between the general and standard forms, and see the graph of the conic section and other important information about it.
Start by setting the values of the coefficients of the general form of the equation, either by typing them in directly or by using the controls.
When enough information to determine a conic section has been entered, its graph will be displayed in the coordinate system, …
… and the standard form of the equation will also be shown, along with information about the conic section.
Once displayed, you may alter the parameters in the standard form, either by typing them in or by using a control.
You could find that you haven’ t entered information that leads to one of the previously mentioned conic sections, and find yourself looking at a line or two, or a point, or even nothing at all!
In this case, you have generated a degenerate conic. Alter some or all of the coefficients until you see a parabola, circle, ellipse, or hyperbola.
It may also be that the conic is not degenerate, but rather that the viewing window needs to be adjusted.
In the lower right of the graphing area are controls to adjust the viewing window.
The arrow controls move the window through which the graph and coordinate system are viewed in the direction indicated by the arrow.
The + and - controls zoom in and out to show less or more of the graphing area, and the zero control resets the zoom level.
If you get lost in the coordinate grid, you can zoom out until you see the origin, use the arrow buttons to center the origin, then zoom in or use the zero button to get a picture closer to what you may be expecting.
See if you can predict which conic sections are generated from a particular set of coefficients before entering those coefficients to check your prediction.
Here’ s a helpful hint: the type of conic section relies on only two terms!