Conic

Definition

A conic is a curve obtained by intersecting a plane with a double-napped cone. The types of conics include ellipses, hyperbolas, and parabolas.

5 Must Know Facts For Your Next Test

1. A circle is a special case of an ellipse where the eccentricity is zero.
2. The standard form of an ellipse's equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
3. The standard form of a hyperbola’s equation is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.
4. In polar coordinates, the general equation for conics can be expressed as $r = \frac{ed}{1 + e\cos(\theta)}$ where $e$ is the eccentricity and $d$ is the directrix.
5. The eccentricity ($e$) determines the shape of the conic: if $0 < e < 1$, it’s an ellipse; if $e = 1$, it’s a parabola; if $e > 1$, it’s a hyperbola.

Review Questions

• What are the different types of conic sections and how are they derived?
• What is the standard form equation of an ellipse and how does it differ from that of a hyperbola?
• How do you express conics in polar coordinates?

Related terms

Ellipse: A type of conic section that appears as an oval-shaped curve, defined by its semi-major and semi-minor axes.

Hyperbola: A type of conic section formed by intersecting a plane with both nappes of the cone, characterized by two separate curves called branches.

Eccentricity: A parameter associated with conic sections that describes their shape; denoted as $e$, it helps classify whether the conic is an ellipse, parabola, or hyperbola.