Answer:
(a) [tex]x ^ 2 + y ^ 2 = 6 ^ 2[/tex] circle centered on the point (0,0) and with a constant radius [tex]r = 6[/tex].
(b) [tex](x-1) ^ 2 + y ^ 2 = 1[/tex] circle centered on the point (1, 0) and with radio [tex]r=1[/tex]
Step-by-step explanation:
Remember that to convert from polar to rectangular coordinates you must use the relationship:
[tex]x = rcos(\theta)[/tex]
[tex]y = rsin(\theta)[/tex]
[tex]x ^ 2 + y ^ 2 = r ^ 2[/tex]
In this case we have the following equations in polar coordinates.
(a) [tex]r = 6[/tex].
Note that in this equation the radius is constant, it does not depend on [tex]\theta[/tex].
As
[tex]r ^ 2 = x ^ 2 + y ^ 2[/tex]
Then we replace the value of the radius in the equation and we have to::
[tex]x ^ 2 + y ^ 2 = 6 ^ 2[/tex]
Then [tex]r = 6[/tex] in rectangular coordinates is a circle centered on the point (0,0) and with a constant radius [tex]r = 6[/tex].
(b) [tex]r = 2cos(\theta)[/tex]
The radius is not constant, the radius depends on [tex]\theta[/tex].
To convert this equation to rectangular coordinates we write
[tex]r = 2cos(\theta)[/tex] Multiply both sides of the equality by r.
[tex]r ^ 2 = 2 *rcos(\theta)[/tex] remember that [tex]x = rcos(\theta)[/tex], then:
[tex]r ^ 2 = 2x[/tex] remember that [tex]x ^ 2 + y ^ 2 = r ^ 2[/tex], then:
[tex]x ^ 2 + y ^ 2 = 2x[/tex] Simplify the expression.
[tex]x ^ 2 -2x + y ^ 2 = 0[/tex] Complete the square.
[tex]x ^ 2 -2x + 1 + y ^ 2 = 1[/tex]
[tex](x-1) ^ 2 + y ^ 2 = 1[/tex] It is a circle centered on the point (1, 0) and with radio [tex]r=1[/tex]
