11.8 Polar Equations of Conic Sections (skip 11.7)

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Presentation transcript:

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. 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

Polar Equation Directrix (d > 0 always) Axis To determine vertices, let: x = – dHorizontalθ = 0 and π x = dHorizontalθ = 0 and π y = – dVertical y = dVertical opp of directrix values that give trig function = 1

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

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

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! V V C F 1 focus ALWAYS at pole!  ellipse to find d: center halfway between vertices

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