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11.8 Polar Equations of Conic Sections

11.8 Polar Equations of Conic Sections. (skip 11.7). We know from Chapter 10, x 2 + y 2 = 64 is an equation of a circle and can be written as r = 8 in polar form.

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11.8 Polar Equations of Conic Sections

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  1. 11.8 Polar Equations of Conic Sections (skip 11.7)

  2. We know from Chapter 10, x2 + y2 = 64 is an equation of a circle and can be written as r = 8 in polar form. We can write polar forms of equations for parabolas, ellipses, & hyperbolas (not circles) that have a focus at the pole and a directrix parallel or perpendicular to the polar axis. * d is directrix and e is eccentricity (we can derive equations using a general definition) * if e = 1  parabola, if e > 1  hyperbola, if 0 < e < 1  ellipse Often useful to use a vertex for the point

  3. values that give trig function = 1 opp of directrix

  4. Ex 1) Identify the conic with equation . Find vertices & graph. ed = 6 2d = 6 d = 3 e = 2 directrix: x = –3 • hyperbola let θ = 0: let θ = π: (–6, 0) (2, π) Note: directrix not necessarily in middle 1 focus ALWAYS at pole!

  5. Try on your own: Ex 2) Identify the conic with equation . Find vertices & graph. ed = ½ 1·d = ½ d = ½ e = 1 directrix: y = ½ parabola let θ = : let θ = : 1 –1 1 focus ALWAYS at pole!

  6. 0 *We can also work backwards to find an equation. Ex 3) Find a polar equation for the conic with the given characteristic. a) Focus at the pole; directrix: y = –3; eccentricity: form: b) The vertices are (2, 0) and (8, π). Find eccentricity & identify the conic. draw a sketch! to find d: V C V F center halfway between vertices 1 focus ALWAYS at pole!  ellipse

  7. Homework #1109  Pg 586 #1, 5, 9, 13, 15, 17, 23, 25, 27, 29

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