Mathlets: Conic Sections

Conic Sections

How to use

Click the "+" and "-" buttons under each value to change that value. Holding a button down causes the action to be repeated.
The "Circle" button sets the coefficients to represent the equation x²+y²-1=0 (the initial values).
The "Hyperbola" button sets the coefficients to represent the equation x²-y²-1=0.
The "Parabola" button sets the coefficients to represent the equation x²-y=0.

Examples

Translated parabola: x²-y=0, translated to (h,k)=(2,1)
Ellipse x²+4y²-4=0
Rotated ellipse: previous example, rotated to t=30
Equation xy-1=0 as rotated hyperbola

Other Notes
The values of h and k give horizontal and vertical (resp.) translation distances, and t gives rotation angle (measured in degrees). Notice how changes in these transformation values affect the coefficients, and how changes in the coefficients affect the transformations.

The lines shown in green in the graph are the following key lines for the conic sections: the major and minor axes for ellipses (crossing at the center of the ellipse), the axis of symmetry and perpendicular line through the vertex for a parabola (crossing at the vertex), and the two perpendicular axes of symmetry (crossing through the center point) for a hyperbola. In all cases, the two lines cross at the point (h,k), and are rotated from the position parallel to the coordinate axes by t degrees. In graphs of hyperbolas, the asymptotes of the hyperbola are shown as orange lines.

The "Type:" label displays what type of conic section is shown in the graph. This can be determined by the value of the discriminant B²-4AC:

If B²-4AC>0, then the graph is a hyperbola.
If B²-4AC=0, then the graph is a parabola.
If B²-4AC<0, then the graph is an ellipse (if B=0 and A=C in this case, then the graph is a circle)

One other important formula determines the relationship between the coefficients and the angle of rotation: tan(2t)=B/(A-C). Note that rotation has no effect on the values of the coefficients D, E, and F, and that t=0 (no rotation) if and only if B=0. The values of the coordinates of the point (h,k) are best determined from the coefficients by first reversing the effect of the rotation (so that B=0), then completing the squares.