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Chapter 9: Q. 29 (page 772)

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22鈥�31.
$r = \frac{4}{1 + \sin 胃}$

Short Answer

Expert verified

The equation is $y = - \frac{1}{8} x^{2} + 2 .$

Step by step solution

01

Step 1. Given information.

The given equation in polar form is $r = \frac{4}{1 + \sin 胃}$ .

02

Step 2. Conversion.

On converting the polar-co-ordinates into cartesian co-ordinates,

$r = \frac{4}{1 + \sin 胃} r + r \sin 胃 = 4 \sqrt{x^{2} + y^{2}} + r 路 \frac{y}{r} = 4 [since r = \sqrt{x^{2} + y^{2}} and r \sin 胃 = y] \sqrt{x^{2} + y^{2}} + y = 4$

03

Step 3. Final answer.

On simplifying the equation,

$\sqrt{x^{2} + y^{2}} + y - y = 4 - y \sqrt{x^{2} + y^{2}} = 4 - y {(\sqrt{x^{2} + y^{2}})}^{2} = (4 - y)^{2} x^{2} + y^{2} = 16 + y^{2} - 8 y x^{2} + y^{2} - y^{2} - 16 = 16 + y^{2} - 8 y - y^{2} - 16 x^{2} - 16 = - 8 y \frac{- 8 y}{- 8} = \frac{x^{2} - 16}{- 8} y = - \frac{1}{8} x^{2} + 2$

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Most popular questions from this chapter

Each of the integrals or integral expressions in Exercises $38 - 44$ represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.

In Exercises 48鈥�55 convert the equations given in rectangular coordinates to equations in polar coordinates.

$y = \sqrt{3} x$

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$r^{2} = \sin 胃$

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

The limacon $r = 2 + \cos 胃$

Given two points in a plane, called _______, a hyperbola is the set of points in the plane for which ________.

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