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12.5 Conic Sections in Polar Coordinates

3 min read•june 24, 2024

in offer a fresh perspective on these familiar shapes. By using distance from a point and an angle, we can describe circles, ellipses, parabolas, and hyperbolas with elegant simplicity. This approach reveals hidden symmetries and relationships.

The of conics connects the shape's , , and in a single equation. This powerful representation allows us to graph and analyze these curves more easily, providing insights into their geometric properties and real-world applications.

Conic Sections in Polar Coordinates

Classification of polar conics

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  • of a conic section in polar coordinates: r=ep1±ecos⁡θr=\frac{ep}{1\pm e\cos\theta}
    • ee represents the which determines the shape of the conic (, , , or )
    • pp is the which is the distance from the (origin) to the (a line perpendicular to the )
  • Classifying conics based on eccentricity (ee)
    • Circle: e=0e=0
      • : r=ar=a, where aa is the radius resulting in a constant distance from the origin
    • Ellipse: 0<e<10<e<1
      • : r=ep1−ecos⁡θr=\frac{ep}{1-e\cos\theta} producing a closed curve with two focal points ()
    • : e=1e=1
      • Polar equation: r=p1−cos⁡θr=\frac{p}{1-\cos\theta} forming an open curve with a single focus and directrix
    • Hyperbola: e>1e>1
      • Polar equation: r=ep1−ecos⁡θr=\frac{ep}{1-e\cos\theta} creating two separate open curves () with two

Graphing conics in polar coordinates

  • Plotting points using polar coordinates (r,θ)(r,\theta)
    • rr represents the from the origin () to the point on the conic
    • θ\theta represents the measured counterclockwise from the polar axis to the line segment connecting the origin to the point
  • Graphing conic sections
    1. Substitute values for θ\theta (usually in the range 0≤θ<2π0\leq\theta<2\pi) into the polar equation
    2. Calculate corresponding rr values for each θ\theta
    3. Plot points (r,θ)(r,\theta) in the polar coordinate system
    4. Connect the points smoothly to form the of the conic section
  • in conic sections
    • Ellipses and hyperbolas exhibit symmetry about the polar axis due to the cos⁡θ\cos\theta term in their equations
    • Parabolas have about the line θ=π2\theta=\frac{\pi}{2} (vertical line) passing through the focus at the origin

Focus, directrix, and eccentricity in polar form

  • Focus
    • The point (origin) from which the distance to any point on the conic is proportional to the distance from that point to the directrix
    • In polar coordinates, the focus is always located at the pole (0,0)(0,0)
  • Directrix
    • A line perpendicular to the polar axis that does not pass through the focus
    • The distance from the focus (origin) to the directrix is the focal parameter (pp)
  • Eccentricity (ee)
    • Determines the shape of the conic section (circle, ellipse, parabola, or hyperbola)
    • Defined as the ratio: e=distance from point to focusdistance from point to directrixe=\frac{\text{distance from point to focus}}{\text{distance from point to directrix}}
    • Eccentricity values: circle (e=0e=0), ellipse (0<e<10<e<1), parabola (e=1e=1), hyperbola (e>1e>1)
  • Relationship between focus, directrix, and eccentricity
    • The eccentricity (ee) determines the type of conic section based on its value relative to 0 and 1
    • The focal parameter (pp) relates the focus (origin) and directrix, appearing in the polar equation of the conic
    • The polar equation r=ep1±ecos⁡θr=\frac{ep}{1\pm e\cos\theta} incorporates the focus (origin), directrix (via pp), and eccentricity (ee) to define the conic section

Polar Form of Conic Sections

  • Polar form represents conic sections using polar coordinates (r,θ)(r,\theta)
  • The polar equation of a conic section relates the radial distance rr to the angular coordinate θ\theta
  • Polar equations provide an alternative way to describe and analyze conic sections, often simplifying certain geometric properties

Key Terms to Review (39)

Angular Coordinate: The angular coordinate, also known as the polar angle, is a key concept in polar coordinate systems. It represents the angle between a reference direction, typically the positive x-axis, and the line connecting the origin to a point on the plane.
Area of a circle: The area of a circle is the amount of space enclosed within its circumference. It is calculated using the formula $A = \pi r^2$, where $r$ is the radius of the circle.
Axes of symmetry: Axes of symmetry are lines that divide a figure into two mirror-image halves. In hyperbolas, these axes typically refer to the transverse and conjugate axes.
Branches: Branches refer to the distinct paths or directions that a mathematical object, such as a function or a curve, can take within a specific context. In the realm of conic sections in polar coordinates, branches describe the various segments or parts that make up the overall shape of a conic section when represented in polar form.
Center of a hyperbola: The center of a hyperbola is the midpoint of the line segment joining its two foci. It is also the point where the transverse and conjugate axes intersect.
Center of an ellipse: The center of an ellipse is the midpoint of both the major and minor axes, serving as the point of symmetry for the ellipse. It is typically denoted by a coordinate pair $(h, k)$ in the Cartesian plane.
Circle: A circle is a closed, two-dimensional shape formed by a set of points that are all equidistant from a fixed point called the center. Circles are fundamental geometric shapes with numerous applications in mathematics, science, and everyday life.
Co-vertices: Co-vertices are the endpoints of the minor axis of an ellipse. They lie on the line segment perpendicular to the major axis, equidistant from the center.
Conic Sections: Conic sections are the curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola, and they have important applications in various fields, including mathematics, physics, and engineering.
Degenerate conic sections: Degenerate conic sections are special cases of conic sections that do not form the usual shapes like ellipses, parabolas, or hyperbolas. They occur when the plane intersects the cone at its vertex or in other ways that produce a single point, a line, or intersecting lines.
Directrix: A directrix is a fixed line used in the geometric definition of a conic section. For a parabola, it is equidistant from any point on the curve to the focus and the directrix.
Directrix: The directrix is a fixed line used in the definition of a conic section, such as an ellipse, hyperbola, or parabola. It serves as a reference point in the geometric construction and mathematical description of these curves.
Eccentricity: Eccentricity measures the deviation of a conic section from being circular. It is denoted by $e$ and determines the shape of the conic.
Eccentricity: Eccentricity is a measure of the shape or deviation of a conic section from a perfect circle. It is a dimensionless quantity that describes the elongation or flattening of a conic section, such as an ellipse, hyperbola, or parabola, and is a fundamental property that characterizes these geometric shapes.
Ellipse: An ellipse is a closed, two-dimensional shape that resembles an elongated circle. It is one of the fundamental conic sections, which are the shapes formed by the intersection of a plane and a cone.
Focal Parameter: The focal parameter is a key characteristic of conic sections when represented in polar coordinates. It defines the distance between the focus and the directrix of the conic, and it is a crucial parameter in determining the shape and orientation of the conic section.
Foci: Foci are two fixed points on the interior of an ellipse or hyperbola used to define and construct these shapes. The sum of the distances from any point on an ellipse to the foci is constant, while the difference of the distances from any point on a hyperbola to the foci is constant.
Foci: The foci of a conic section, such as an ellipse or hyperbola, are the points around which the shape is constructed. They are the focal points that define the shape and properties of the conic section.
Focus: A focus (plural: foci) is a point used to define and describe conic sections such as ellipses, parabolas, and hyperbolas. In these shapes, distances to the focus have special geometric properties.
Focus: The focal point or point of concentration, the center of interest or activity. In the context of conic sections, the focus refers to a specific point that defines the shape and properties of these geometric figures.
General Form: The general form of an equation is a standardized way of expressing the equation that reveals its underlying structure and characteristics. This term is particularly relevant in the context of various mathematical functions and conic sections, as it allows for a concise and informative representation of these entities.
Hyperbola: A hyperbola is a type of conic section, which is a curve formed by the intersection of a plane and a cone. It is characterized by two symmetric branches that open in opposite directions and are connected by a center point.
Latus rectum: The latus rectum of a parabola is the line segment perpendicular to the axis of symmetry that passes through the focus and whose endpoints lie on the parabola. It is used to describe certain geometric properties of parabolas.
Latus Rectum: The latus rectum is a line segment that passes through the focus of a conic section and is perpendicular to the major axis. It is an important geometric property that helps define the shape and characteristics of ellipses, hyperbolas, and parabolas.
Parabola: A parabola is a symmetric curve that represents the graph of a quadratic function. It can open upward or downward depending on the sign of the quadratic coefficient.
Parabola: A parabola is a curved, U-shaped line that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola, and has many important applications in mathematics, science, and engineering.
Polar Axis: The polar axis is the reference line or axis used to define the position of points in a polar coordinate system. It serves as the starting point for measuring the angle, known as the polar angle, in the polar coordinate plane.
Polar Coordinates: Polar coordinates are a system of representing points in a plane using the distance from a fixed point (the pole) and the angle from a fixed direction (the polar axis). This system provides an alternative to the Cartesian coordinate system and is particularly useful for describing circular and periodic phenomena.
Polar equation: A polar equation is a mathematical expression that defines a relationship between the radius $r$ and the angle $\theta$ of a point in the polar coordinate system. It is commonly used to describe conic sections and other geometric shapes.
Polar Equation: A polar equation is a mathematical expression that defines a curve or shape in polar coordinates, where the position of a point is specified by its distance from a fixed origin (the pole) and the angle it forms with a fixed reference direction (the polar axis).
Polar Form: Polar form is a way of representing complex numbers and graphing equations in a polar coordinate system. It involves expressing a complex number or a curve in terms of its magnitude (or modulus) and angle (or argument) rather than its rectangular (Cartesian) coordinates.
Polar form of a conic: The polar form of a conic is an equation representing conic sections (ellipse, parabola, hyperbola) using polar coordinates $(r, \theta)$. It often involves parameters like the eccentricity $e$ and the directrix.
Polar Graph: A polar graph is a graphical representation of a function where the independent variable is the angle, measured in radians, and the dependent variable is the distance from the origin. Polar graphs are particularly useful for visualizing and analyzing functions that are more naturally expressed in polar coordinates rather than rectangular coordinates.
Pole: The pole is a special point in a polar coordinate system that serves as the origin, around which all other points are defined by their distance and angle. It is the fixed reference point from which the coordinates of any other point in the plane are measured.
R = ep / (1 ± e cos θ): 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).
Radial Distance: Radial distance, in the context of polar coordinates and conic sections, refers to the distance from the origin (or pole) to a point on a curve or graph. It represents the magnitude or length of the vector from the origin to the point, and is a crucial component in describing the position and shape of objects in polar coordinate systems.
Reflectional Symmetry: Reflectional symmetry, also known as line symmetry or mirror symmetry, is a type of symmetry where a shape or object can be divided into two equal halves that are mirror images of each other along a specific line or axis. This means that if the shape or object is folded along this line, the two halves will perfectly align and match each other.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis. It is a fundamental concept in mathematics and geometry that describes the balanced and harmonious arrangement of elements in an object or function.
Vertices: Vertices refer to the specific points on a conic section that represent the maximum or minimum values of the function. They are the most critical points that define the shape and orientation of the conic section.
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