Conic section

Definition

A conic section is a curve obtained by intersecting a cone with a plane. The four types are circles, ellipses, parabolas, and hyperbolas.

5 Must Know Facts For Your Next Test

1. The general equation of a conic section in rotated axes can be written as $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
2. Rotation of axes can eliminate the $Bxy$ term, simplifying the conic section's equation.
3. To rotate the axes to eliminate the $Bxy$ term, use the angle $\theta$ where $\tan(2\theta) = \frac{B}{A - C}$.
4. A hyperbola is formed when the plane intersects both nappes of the cone.
5. In analytic geometry, completing the square is often used to rewrite conic sections in their standard forms.

Review Questions

• What happens to the $Bxy$ term when you rotate the axes correctly?
• How do you determine the angle of rotation for eliminating the $Bxy$ term in a conic section's equation?
• What are the four types of conic sections?

Related terms

Hyperbola: A type of conic section formed by intersecting a double-napped cone with a plane that cuts through both nappes.

Ellipse: A type of conic section formed by intersecting a cone with a plane that cuts through only one nappe but does not pass through its base.

Parabola: A type of conic section formed by intersecting a cone with a plane parallel to one side (a generator) of the cone.