Conic section

Definition

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

5 Must Know Facts For Your Next Test

1. Conic sections can be expressed in parametric form.
2. The general quadratic equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ represents all conic sections.
3. Ellipses have two foci, parabolas have one focus, and hyperbolas have two foci.
4. In polar coordinates, conic sections have the form $r = \frac{ed}{1 - e\cos(\theta)}$ where $e$ is the eccentricity.
5. The eccentricity ($e$) determines the type of conic: $e=0$ (circle), $0<e<1$ (ellipse), $e=1$ (parabola), and $e>1$ (hyperbola).

Review Questions

• How can you represent a conic section using parametric equations?
• What is the significance of the eccentricity in determining the type of conic section?
• Write down the general quadratic equation that represents all conic sections.

Related terms

Eccentricity:

A parameter that determines the shape of a conic section. It is denoted by $e$.

Focus:

A point used to define and construct a conic section.

Directrix:

A line used in conjunction with a focus to define a conic section.