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

Question

asked Aug 8, 2020 164k views

1 Answer

WeirdlyCheezy · Answer 1 · 2020-08-13T21:04:37+0000

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)
.

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