Conic sections can be described in polar coordinates by placing one focus at the pole and measuring each point with a distance r and an angle theta. This form is powerful because ellipses, parabolas, and hyperbolas all come from one focus-directrix rule. It is especially useful in orbital motion, where the Sun or another central body often sits at a focus.
The key parameter is eccentricity, which tells how stretched the conic is and what type of curve appears.
Key Facts
- Focus-directrix definition: eccentricity e = distance to focus / distance to directrix.
- Standard polar conic form: r = ed / (1 ± e cos theta) or r = ed / (1 ± e sin theta).
- If 0 < e < 1, the conic is an ellipse.
- If e = 1, the conic is a parabola.
- If e > 1, the conic is a hyperbola.
- For r = ed / (1 + e cos theta), the directrix is x = d and the axis is horizontal.
Vocabulary
- Conic section
- A curve formed by slicing a cone, including circles, ellipses, parabolas, and hyperbolas.
- Polar coordinates
- A coordinate system that locates a point using its distance r from the pole and its angle theta from a reference ray.
- Focus
- A fixed point used to define a conic by comparing distances from the curve to the focus and to a directrix.
- Directrix
- A fixed line used with a focus to define a conic through a constant distance ratio.
- Eccentricity
- The constant ratio e that measures how far a conic differs from a circle and determines whether it is an ellipse, parabola, or hyperbola.
Common Mistakes to Avoid
- Confusing e with ed is wrong because e is the eccentricity, while ed is the numerator containing both eccentricity and the directrix distance.
- Classifying by the numerator instead of by e is wrong because the type of conic depends only on eccentricity: e < 1, e = 1, or e > 1.
- Ignoring the sign and trig function in the denominator is wrong because cos theta versus sin theta and plus versus minus determine the axis direction and directrix placement.
- Assuming r is always positive is wrong because polar equations can produce negative r values, which plot in the opposite direction from the given angle.
Practice Questions
- 1 Identify the conic and find e for r = 12 / (1 + 0.5 cos theta).
- 2 For r = 8 / (1 - 2 sin theta), identify the conic and compute the value of d if the equation is written as r = ed / (1 - e sin theta).
- 3 Explain how the graph changes when the eccentricity in r = ed / (1 + e cos theta) increases from 0.6 to 1 to 1.4 while d stays positive.