Directrix

Definition

The directrix is a fixed, straight line that, along with the focus, defines the shape and position of a conic section, such as a parabola, ellipse, or hyperbola. It serves as a reference point for the curve and is used in the mathematical equations that describe these geometric shapes.

5 Must Know Facts For Your Next Test

1. The directrix is a fixed, straight line that, along with the focus, defines the shape and position of a conic section.
2. The distance between the focus and the directrix determines the eccentricity of the conic section, which in turn affects the shape of the curve.
3. The vertex of a conic section is the point that is closest to the directrix and is where the curve changes direction.
4. The equation of a parabola includes the distance between the focus and the directrix, which is used to determine the shape and position of the curve.
5. In the equation of an ellipse or hyperbola, the distance between the focus and the directrix is used to calculate the eccentricity of the curve.

Review Questions

• Explain the role of the directrix in the graph of a quadratic function.
  • The directrix is a fixed, straight line that, along with the focus, defines the shape and position of a parabola, which is the graph of a quadratic function. The distance between the focus and the directrix determines the eccentricity of the parabola, which in turn affects the shape and orientation of the curve. The vertex of the parabola is the point that is closest to the directrix, and the equation of the parabola includes the distance between the focus and the directrix as a key parameter.
• Describe the relationship between the directrix and the focus in the equation of an ellipse or hyperbola.
  • In the equation of an ellipse or hyperbola, the distance between the focus and the directrix is used to calculate the eccentricity of the curve. The eccentricity, in turn, determines the shape and elongation of the ellipse or hyperbola. The focus and directrix are both essential in defining the geometric properties of these conic sections, as they provide the necessary information to determine the equation and graph of the curve.
• Analyze how changes in the position or orientation of the directrix would affect the graph of a conic section.
  • $$\text{If the position or orientation of the directrix is changed, it will result in a corresponding change in the shape, size, and position of the conic section. For example, moving the directrix closer to or farther away from the focus will affect the eccentricity of the curve, making it more or less elongated. Rotating the directrix will change the orientation of the conic section. These changes to the directrix are directly reflected in the equation of the curve and, consequently, its graphical representation. Understanding the relationship between the directrix and the other defining elements of a conic section is crucial for accurately graphing and analyzing these important geometric shapes.}$$

Related terms

Focus:

The focus is a fixed point that, along with the directrix, defines the shape and position of a conic section. It is a key element in the equations that describe the curve.

Eccentricity:

Eccentricity is a measure of how elongated or flattened a conic section is, and it is related to the distance between the focus and the directrix.

Vertex:

The vertex is the point on a conic section that is closest to the directrix and is the point where the curve changes direction.