Introduction
In polar coordinates, all conic sections can be expressed with a single unified formula. The type of conic depends entirely on the eccentricity e.
Prerequisite Connection
You understand rotation of conics and can work with discriminants.
Today's Increment
We learn the unified polar form for all conics and how eccentricity determines the type.
Why This Matters
Kepler's Laws use polar conic equations to describe planetary orbits around the Sun.
Polar Form of Conics
Unified Polar Form (focus at pole)
Ellipse
Parabola
Hyperbola
Parameters
- • e = eccentricity (determines conic type)
- • d = distance from focus to directrix
- • ± cos θ = directrix vertical (left/right of pole)
- • ± sin θ = directrix horizontal (above/below pole)
Interactive: See How Eccentricity Changes the Conic
Worked Examples
Example 1: Identify the Conic
Identify:
Divide by 2:
So e = 0.5, ed = 3, d = 6.
Since e = 0.5 < 1, this is an ellipse.
Example 2: Write Polar Equation
Write polar equation for parabola with directrix x = -3.
Parabola: e = 1. Directrix left of pole: use (1 − cos θ).
d = 3, so ed = 3.
Example 3: Hyperbola
Identify:
e = 2, ed = 4, d = 2. Directrix below pole (−sin).
Since e = 2 > 1, this is a hyperbola.
Common Pitfalls
Not putting in standard form - Divide so denominator starts with 1.
Confusing ± signs - Sign determines directrix location.
Real-World Application
Planetary Orbits
Planets orbit the Sun in ellipses with the Sun at one focus. Kepler's laws use polar conic equations to describe orbital paths.
Practice Quiz
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