Polar Equation of Conics -- D eo

An Alternative Definition of Conics Let L be a fixed line (the directrix); let F be a fixed point (the focus) not on L; and let e be a positive constant ( the eccentricity). A conic is the set of all points P in the plane such that PF / PL = e where PF is the distance from P to F and PL is the distance from P to L. The conic is a parabola if e = 1, an ellipse if e ＜ 1, and a hyperbola if e ＞ 1.

For an ellipse and a hyperbola, the eccentricity e is given by e = c / a, where c is the distance from the center to a focus and a is the distance from the center to a vertex. Eccentricity

Polar Equation of Conics A polar equation of any of the four forms r = ep / (1 ± e cos θ), r = ep / (1 ± e sin θ) is a conic section. The conic is a parabola if e = 1, an ellipse if 0 ＜ e ＜ 1, and a hyperbola if e ＞ 1.

EquationDescription r = ep / (1 + e cos θ) Vertical directrix p units to the right of the pole(or focus) r = ep / (1 - e cos θ) Vertical directrix p units to the left of the pole(or focus) r = ep / (1 + e sin θ) Horizontal directrix p units above the pole(or focus) r = ep / (1 - e sin θ) Horizontal directrix p units below the pole(or focus)

Example r = 1 / (1 + cos θ) a. Parabola b. vertical, 1 unit to the right of the pole c. vertex (1/2, 0)

Converting from Polar Equations to Rectangular Equations Convert to a rectangular equation: r = 2/(1-sinθ) r = 2 / (1- sin θ) r - r sin θ = 2 r = r sin θ + 2 √x² + y² = y + 2 x² + y² = y² + 4y +4 x² - 4y - 4 = 0

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