Calculus Volume 3 · free preview
Conic Sections
Parabolas, ellipses, and hyperbolas are the conic sections, defined by eccentricity and focus-directrix geometry, unified by the polar equation r equals ed over 1 plus e cos theta with the eccentricity discriminating the type.
1. The conic family
The conic family
2. Reflective and orbital properties
Reflective and orbital properties
3. The polar conic equation
The polar conic equation
4. Graphing and standard forms
Graphing and standard forms
5. Eccentricity discrimination
The eccentricity e discriminates the conic families. Which value gives a parabola?
6. Ellipse focus relation
For the ellipse x^2/a^2 + y^2/b^2 = 1 with a greater than b, the foci satisfy which relation?
7. Parabola reflection property
A parabolic reflector works by which geometric property?
8. Polar conic equation
The polar equation r = ed/(1 + e cos theta) with e = 1/2 describes which conic?
9. Kepler first law
Kepler's first law states that planets move in which conic with the sun at a focus?
10. Perihelion distance
For the ellipse r = ed/(1 + e cos theta), the perihelion occurs at theta = pi. As an expression, what is r there? Answer like ed/(1 + e).
11. Hyperbola asymptote condition
For e greater than 1 the denominator 1 + e cos theta vanishes at cos theta = -1/e, giving the hyperbola's what? Answer with one word.
12. Discriminant of conic
The general quadratic's discriminant B^2 - 4AC classifies conics. For the ellipse it is which sign? Answer with one word.
13. Discriminant sign summary
The conic discriminant B^2 - 4AC is zero for the parabola, positive for the hyperbola, and which sign for the ellipse? Answer with one word.
14. Directrix location
In the polar conic r = ed/(1 + e cos theta), the directrix sits at x equals which value? Answer with a single letter.
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