SECTION 10.6 Polar Equations of Conics 733 2 Convert the Polar Equation of a Conic to a Rectangular Equation ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 10.6 Assess Your Understanding 1. If x y , ( ) are the rectangular coordinates of a point P and r, θ ( ) are its polar coordinates, then x = and y = . (pp. 60 0–607) 2. Transform the equation r 6cosθ = from polar coordinates to rectangular coordinates. (pp. 600–607) Converting a Polar Equation to a Rectangular Equation Convert the polar equation r 1 3 3cosθ = − to a rectangular equation. Solution EXAMPLE 4 The strategy here is to rearrange the equation and square both sides before converting the equation in polar coordinates to an equation in rectangular coordinates. r r r r r r r x y x x y x x y x 1 3 3cos 3 3 cos 1 3 1 3 cos 9 1 3 cos 9 1 3 9 9 9 6 1 9 6 1 6 2 2 2 2 2 2 2 2 2 θ θ θ θ ( ) ( ) ( ) ( ) = − − = = + = + + = + + = + + = + This is the equation of a parabola in rectangular coordinates. Rearrange the equation. Square both sides. x y r x r ; cos 2 2 2 θ + = = Now Work PROBLEM 25 Skill Building In Problems 7–12, identify the conic defined by each polar equation. Also give the position of the directrix. 7. r 1 1 cosθ = + 8. r 3 1 sinθ = − 9. r 4 2 3sinθ = − 10. r 2 1 2cosθ = + 11. r 3 4 2 cosθ = − 12. r 6 8 2 sinθ = + Concepts and Vocabulary 3. A is the set of points P in a plane for which the ratio of the distance from a fixed point F, called the , to P to the distance from a fixed line D, called the , to P equals a constant e. 4. A conic has eccentricity e. If e 1, = the conic is a(n) . If e 1, < the conic is a(n) . If e 1, > the conic is a(n) . 5. Multiple Choice If r, θ ( ) are polar coordinates, the equation r 2 2 3sinθ = + defines a(an) . (a) parabola (b) hyperbola (c) ellipse (d) circle 6. True or False The eccentricity e of an ellipse is c a , where a is the distance of a vertex from the center and c is the distance of a focus from the center. 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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