R = ep / (1 ± e cos θ)

Definition

The equation r = ep / (1 ± e cos θ) is a fundamental expression in the study of conic sections in polar coordinates. It describes the relationship between the polar coordinates (r, θ) and the parameters of a conic section, specifically the eccentricity (e) and the semi-latus rectum (p).

5 Must Know Facts For Your Next Test

1. The equation r = ep / (1 ± e cos θ) is used to describe the shape and size of conic sections in polar coordinates.
2. The parameter 'p' represents the semi-latus rectum, which is the distance from the focus to the curve along a line perpendicular to the major axis.
3. The parameter 'e' represents the eccentricity of the conic section, which determines whether the curve is an ellipse, parabola, or hyperbola.
4. The '±' in the equation indicates that the curve can open in either the positive or negative direction, depending on the sign of the eccentricity.
5. Understanding this equation is crucial for analyzing the properties and characteristics of conic sections in polar coordinate systems.

Review Questions

• Explain how the equation r = ep / (1 ± e cos θ) relates to the properties of conic sections.
  • The equation r = ep / (1 ± e cos θ) directly connects the polar coordinates (r, θ) to the defining parameters of a conic section: the eccentricity (e) and the semi-latus rectum (p). The eccentricity determines the type of conic section (ellipse, parabola, or hyperbola), while the semi-latus rectum defines the size and shape of the curve. By plugging in different values for e and p, this equation can be used to generate the polar equation of any conic section, allowing for the analysis of its characteristics and behavior.
• Describe how the '±' sign in the equation r = ep / (1 ± e cos θ) affects the orientation of the conic section.
  • The '±' sign in the equation r = ep / (1 ± e cos θ) indicates that the conic section can open in either the positive or negative direction, depending on the sign of the eccentricity (e). If e is positive, the conic section will open in the positive direction, while if e is negative, the conic section will open in the negative direction. This sign change is crucial for determining the overall orientation of the curve and its relationship to the polar coordinate system.
• Analyze how changes in the eccentricity (e) and semi-latus rectum (p) parameters affect the shape and size of the conic section described by the equation r = ep / (1 ± e cos θ).
  • The eccentricity (e) and semi-latus rectum (p) parameters in the equation r = ep / (1 ± e cos θ) directly influence the shape and size of the resulting conic section. As the eccentricity (e) increases from 0 to 1, the conic section transitions from a circle to an ellipse, and then to a parabola. If the eccentricity is greater than 1, the conic section becomes a hyperbola. Meanwhile, the semi-latus rectum (p) determines the overall scale of the curve, with larger values of p resulting in a larger conic section. By manipulating these two parameters, the equation can be used to generate a wide variety of conic sections with different shapes and sizes.

Related terms

Polar Coordinates: A coordinate system that specifies the location of a point on a plane by a distance from a reference point (the pole) and an angle from a reference direction (the polar axis).

Conic Sections: The curves that are formed by the intersection of a plane and a cone, including the circle, ellipse, parabola, and hyperbola.

Eccentricity (e): A measure of how much a conic section deviates from being circular, with a value between 0 and 1 for an ellipse, equal to 1 for a parabola, and greater than 1 for a hyperbola.