Short Answer

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$r = \frac{α}{1 + \sin θ}$

The equation of the parabola is $x^{2} = - 2 y α + α^{2}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given polar equation is $r = \frac{α}{1 + \sin θ}$ .

Step 2. Conversion.

Converting the polar into cartesian co-ordinates,

$r = \frac{α}{1 + \sin θ} r (1 + \sin θ) = α r + r \sin θ = α \sqrt{x^{2} + y^{2}} + r \cdot \frac{y}{r} = α [since r = \sqrt{x^{2} + y^{2}} and r \sin θ = y] \sqrt{x^{2} + y^{2}} + y = α$

Step 3. Final answer.

On simplifying the equation,

$\sqrt{x^{2} + y^{2}} + \ not y - \ not y = α - y \sqrt{x^{2} + y^{2}} = α - y {(\sqrt{x^{2} + y^{2}})}^{2} = (α - y)^{2} x^{2} + y^{2} = α^{2} + y^{2} - 2 y α x^{2} + y^{2} - y^{2} = α^{2} + y^{2} - 2 y α - y^{2} x^{2} = - 2 y α + α^{2}$

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Calculus

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Short Answer

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$r = \frac{α}{1 + \sin θ}$

Step by Step Solution

Step 1. Given information.

Step 2. Conversion.

Step 3. Final answer.

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Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31. r=α1+sinθ

Step by Step Solution

Step 1. Given information.

Step 2. Conversion.

Step 3. Final answer.

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Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22–31.
$r = \frac{α}{1 + \sin θ}$