164k views
3 votes
Transform each polar equation to an equation in rectangular coordinates and identify its shape.

(a) r=6
(b) r= 2 cos theta

please show ur work

1 Answer

6 votes

Answer:

(a)
x ^ 2 + y ^ 2 = 6 ^ 2

(b)
(x-1) ^ 2 + y ^ 2 = 1

Explanation:

Remember that to convert from polar to rectangular coordinates you must use the relationship:


x = rcos(\theta)


y = rsin(\theta)


x ^ 2 + y ^ 2 = r ^ 2

In this case we have the following equations in polar coordinates.

(a)
r = 6
.

Note that in this equation the radius is constant, it does not depend on
\theta.

As
r ^ 2 = x ^ 2 + y ^ 2

Then we replace the value of the radius in the equation and we have to::


x ^ 2 + y ^ 2 = 6 ^ 2

Then
r = 6 in rectangular coordinates is a circle centered on the point (0,0) and with a constant radius
r = 6.

(b)
r = 2cos(\theta)

The radius is not constant, the radius depends on
\theta.

To convert this equation to rectangular coordinates we write


r = 2cos(\theta) Multiply both sides of the equality by r.


r ^ 2 = 2 *rcos(\theta) remember that
x = rcos(\theta), then:


r ^ 2 = 2x remember that
x ^ 2 + y ^ 2 = r ^ 2, then:


x ^ 2 + y ^ 2 = 2x Simplify the expression.


x ^ 2 -2x + y ^ 2 = 0 Complete the square.


x ^ 2 -2x + 1 + y ^ 2 = 1


(x-1) ^ 2 + y ^ 2 = 1 It is a circle centered on the point (1, 0) and with radio
r=1

User WeirdlyCheezy
by
8.6k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories