Hyperbola
Hyperbola
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Conic Sections Defined:
If a point P moves so that its distance from a fixed point (called the focus) divided by its distance from a fixed line (called the directrix) is a constant ε (called the eccentricity), then the curve described by P is called a conic (so-called because such curves can be obtained by intersecting a plane and a cone at different angles).

If the focus is chosen at origin O, the equation of a conic in polar coordinates (r, θ) is, if OQ = p and LM = D,
r = p / (1 - ε cos θ) = (ε D) / (1 - ε cos θ)
conicdefination

The conic is
  1. an ellipse if ε < 1
  2. a parabola if ε = 1
  3. a hyperbola if ε > 1.





The ellipse is a closed curve with two foci located inside the curve. It has major and minor axes equal to the length and width of the ellipse.

The parabola is an open curve. It has one focus and one axis, located on the line drawn between the vertex and the focus.

The hyperbola has two open curves. They are mirror images of each other. There are two asymptotes with slopes = +-(b / a). The hyperbola has two foci, each located inside its own curve. It has major and a minor axes located between the two curves.


Rectangular Coordinates - ellipse and hyperbola with center C(xo, yo) and major axis parallel to the x-axis, parabola with axis parallel to the x-axis.

  1. Ellipse: (x - xo)2 / a2 + (y - yo)2 / b2 = 1, 2a = major axis, 2b = minor axis
  2. Parabola: (y - yo )2 = 4a (x - xo), parabola opening to the right, a is a constant > 0
    (y - yo)2 = -4a (x - xo), parabola opening to the left, a is a constant > 0
  3. Hyperbola: (x - xo)2 / a2 - (y - yo)2 / b2 = 1, 2a = major axis and 2b = minor axis

Polar Coordinates - ellipse and hyperbola with center C(xo, yo) and major axis parallel to the x-axis, parabola with axis parallel to the x-axis.

  1. Ellipse: r2 = (a2 b2) / (a2 sin2 θ + b2 cos2 θ), the center is at the origin, 2a = major axis, 2b = minor axis
    r = (a(1 - ε2)) / (1 - ε cos θ), the center is on the x-axis and one focus is at the origin
  2. Parabola: r = 2a / (1 - cos θ), the focus is at the origin
  3. Hyperbola: r2 = (a2 b2) / (a2 sin2 θ - b2 cos2 θ), the center is at the origin, 2a = major axis, 2b = minor axis
    r = (a(ε2 - 1)) / (1 - ε cos θ), the center is on the x-axis and one focus is at the origin

From Schaum's Outline Series THE MATHEMATICAL HANDBOOK of FORMULAS and TABLES.

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