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Chapter 9: Q 37. (page 731)

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.
$r = 5 \sin 胃$

Short Answer

Expert verified

The required equation is $x^{2} + {(y - \frac{5}{2})}^{2} = {(\frac{5}{2})}^{2}$ .

Step by step solution

01

Step 1. Given information.

The given equation in polar coordinates is:

$r = 5 \sin 胃$

02

Step 2. Find the equation in rectangular coordinates.

$r = 5 \sin 胃$

First, multiply both sides byr,

$r^{2} = 5 r \sin 胃 r^{2} = 5 y [Since r \sin 胃 = y] x^{2} + y^{2} = 5 y [Since r^{2} = x^{2} + y^{2}]$

Now add $- 5 y$ on both sides of the equation,

$x^{2} + y^{2} - 5 y = 5 y - 5 y x^{2} + y^{2} - 5 y = 0$

Complete the square in y,

$x^{2} + {(y - \frac{5}{2})}^{2} - {(\frac{5}{2})}^{2} = 0$

now add ${(\frac{5}{2})}^{2}$ on both sides of the equation,

$x^{2} + {(y - \frac{5}{2})}^{2} - {(\frac{5}{2})}^{2} + {(\frac{5}{2})}^{2} = 0 + {(\frac{5}{2})}^{2} x^{2} + {(y - \frac{5}{2})}^{2} = {(\frac{5}{2})}^{2}$

Therefore, the equation in rectangular coordinates is $x^{2} + {(y - \frac{5}{2})}^{2} = {(\frac{5}{2})}^{2}$ .

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Most popular questions from this chapter

Use polar coordinates to graph the conics in Exercises 44鈥�51.

$r = \frac{2}{4 + \sin 胃}$

Use polar coordinates to graph the conics in Exercises 44鈥�51.

$r = \frac{5}{1 + \cos 胃}$

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

$r = 胃$

Use polar coordinates to graph the conics in Exercises 44鈥�51.

$r = \frac{2}{4 + \cos 胃}$

In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

$(6, 2 \sqrt{3})$

See all solutions

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