POLAR FORM OF CONIC SECTIONS DR SHILDNECK CONIC

POLAR FORM OF CONIC SECTIONS DR. SHILDNECK

CONIC SECTIONS • In general, a conic section can be defined as the locus of points such that the distance from a point, P, to the focus and the distance from a point (P) to a fixed line not containing P (the directrix) is a constant ratio. • The constant ratio is called the eccentricity of the conic and is denoted as e. directrix P Q F

ECCENTRICITY The type of conic section can be determined by finding its eccentricity. The values of e for each type of conic are listed below. Note, e is always positive.

POLAR FORM OF CONICS Directrix x = d • Let our conic have a focus at the origin. P (x, y) d-x • Begin with the definition for e. • Rewrite as a constant multiplier. Q • Find PF and PQ. F (0, 0) • Substitute. • Convert to Polar form using conversion formulas. Begin with Multiply by PQ The distance PQ is the horizontal distance from d to x The distance PQ is the slanted distance from (x, y) to (0, 0) Distribute Add to get r’s on one side Factor out r Divide to solve for r

POLAR EQUATIONS OF CONICS •

EXAMPLES • The key number in this problem is the one in the denominator Factor Simplify and reduce… we want only “ 1+ecos(Ѳ)” in the denominator From this, we can tell that the eccentricity is e = 0. 5. We also know that ed = 3.

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ASSIGNMENT Alternate Text (from Blog) Page 726 • # 5 -10 all • #11 -19 odd • #33 -47 odd