Question

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

Answer 1

(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`