Chapter 9: Q. 31 (page 772)

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

Short Answer

Expert verified

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

Step by step solution

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

Short Answer

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Conversion.

Step 3. Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Logic and Functions

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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