When B ≠ 0, the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents a conic section, just as the equation Ax2 + Cy2 + Dx + Ey + F = 0 does, but without any of its axes parallel to a coordinate axis. Rotating the conic section about the origin by an angle θ changes the coefficients of the equation, but not the shape of the conic section. If θ is chosen so that the conic section has an axis parallel to a coordinate axis, then B = 0 and the equation may be written in standard form.
The standard form of the equation is an approximation
Enter the coefficients in the entry fields below, either by using the controls or by typing them into the spaces. Choose the values A, B, and C as follows to graph a desired conic section:
B2 - 4AC = 0 for a parabola
B2 - 4AC < 0 for an ellipse
B2 - 4AC > 0 for a hyperbola
If you still have trouble graphing a particular conic section after choosing appropriate value for A, B, and C, try adjusting the value of F. (Note: a circle is a special case of an ellipse.)
Once you have the graph of a conic section, adjust the angle of rotation and see how the coefficients of the equation change.
When you need to approximate a coefficient, because it has a square root or for some other reason, try to use at least seven digits in the approximation to minimize roundoff error.
Exploration 1:
For any rotation that makes
B = 0, there others that will do the same. How are these rotation values related?
Exploration 2:
Compare the sums of the values of the coefficients of the squared terms before and after a rotation. In other words, how does A + C compare with A' + C'?
Exploration 3:
The quantity B2 - 4AC is called the discriminant. The sign of the discriminant yields information about the type of conic section that the equation represents. Since rotating a conic section about the origin doesn’t change its shape, the sign of the discriminant doesn’t change, so (B')2 - 4A'C' has the same sign as B2 - 4AC. What else can be said about the value of the discriminant after a rotation?
