12.5 Conic Sections in Polar Coordinates

Conic sections in polar coordinates 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 polar form of conics connects the shape's eccentricity, focus, and directrix 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

• General form of a conic section in polar coordinates: $r=\frac{ep}{1\pm e\cos\theta}$
  • $e$ represents the eccentricity which determines the shape of the conic (circle, ellipse, parabola, or hyperbola)
  • $p$ is the focal parameter which is the distance from the focus (origin) to the directrix (a line perpendicular to the polar axis)
• Classifying conics based on eccentricity ($e$)
  • Circle: $e=0$
    • Polar equation: $r=a$, where $a$ is the radius resulting in a constant distance from the origin
  • Ellipse: $0<e<1$
    • Polar equation: $r=\frac{ep}{1-e\cos\theta}$ producing a closed curve with two focal points (foci)
  • Parabola: $e=1$
    • Polar equation: $r=\frac{p}{1-\cos\theta}$ forming an open curve with a single focus and directrix
  • Hyperbola: $e>1$
    • Polar equation: $r=\frac{ep}{1-e\cos\theta}$ creating two separate open curves (branches) with two foci

Graphing conics in polar coordinates

• Plotting points using polar coordinates $(r,\theta)$
  • $r$ represents the radial distance from the origin (pole) to the point on the conic
  • $\theta$ represents the angular coordinate measured counterclockwise from the polar axis to the line segment connecting the origin to the point
• Graphing conic sections
  1. Substitute values for $\theta$ (usually in the range $0\leq\theta<2\pi$) into the polar equation
  2. Calculate corresponding $r$ values for each $\theta$
  3. Plot points $(r,\theta)$ in the polar coordinate system
  4. Connect the points smoothly to form the polar graph of the conic section
• Symmetry in conic sections
  • Ellipses and hyperbolas exhibit symmetry about the polar axis due to the $\cos\theta$ term in their equations
  • Parabolas have reflectional symmetry about the line $\theta=\frac{\pi}{2}$ (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 $(0,0)$
• 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 ($p$)
• Eccentricity ($e$)
  • Determines the shape of the conic section (circle, ellipse, parabola, or hyperbola)
  • Defined as the ratio: $e=\frac{\text{distance from point to focus}}{\text{distance from point to directrix}}$
  • Eccentricity values: circle ($e=0$), ellipse ($0<e<1$), parabola ($e=1$), hyperbola ($e>1$)
• Relationship between focus, directrix, and eccentricity
  • The eccentricity ($e$) determines the type of conic section based on its value relative to 0 and 1
  • The focal parameter ($p$) relates the focus (origin) and directrix, appearing in the polar equation of the conic
  • The polar equation $r=\frac{ep}{1\pm e\cos\theta}$ incorporates the focus (origin), directrix (via $p$), and eccentricity ($e$) to define the conic section

Polar Form of Conic Sections

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