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Section 8-5

Section 8-5. Polar Equations of Conics. Focus –Directrix Definition. all conics can be defined using a special definition (similar to the parabola definition)

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Section 8-5

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  1. Section 8-5 Polar Equations of Conics

  2. Focus –Directrix Definition • all conics can be defined using a special definition (similar to the parabola definition) • a conic section is the set of all points whose distances from a particular point (focus) and a particular line (the directrix) have a constant ratio • the constant ratio is the eccentricity (e) • if e > 1 hyperbola if e < 1 ellipse if e = 1 parabola

  3. Focus Directrix Defintion • let P be a point on the conic, F is the focus, and D be the closest point on the directrix P D F the directrix

  4. Polar Equation of Conics • the new formula for eccentricity can be used to derive polar equations for conics • place the focus of the conic at (0 , 0) and call the directrix x = k P(r,θ) D F(0,0) x = k

  5. Polar Equation of Conics P(r,θ) D r θ F(0,0) rcos θ x = k

  6. Polar Equations of Conics

  7. Polar Equations of Conics • the formula derived in the previous slide is the polar form for a conic based on the eccentricity (e) and a value k which comes from the directrix • there are three other formulas that are just like it depending on the orientation of the conic in relation to the focus and directrix • the next four slides show the four different formula and their orientation

  8. F(0 , 0) directrix x = k

  9. F (0 , 0) directrix x = – k

  10. directrix y = k F(0 , 0)

  11. F(0 , 0) directrix y = – k

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