Determining Cartesian equation

[vwguy]

How do I determine the catesian equations of the following polar equations:
r=12/2+6cos(theta)
r=3/2-2sin(theta)
r=12/6-2sin(theta-pie/3)

 

[morson]

r=12/2+6cos(theta)

Multiply through by (2 + 6cos#): r(2 + 6cos#) = 12

2r + 6rcos# = 12

Now make use of the identities x = rcos#, r^2 = x^2 + y^2

r=3/2-2sin(theta)

Again, multiply through by (2 - 2sin#): r(2 - 2sin#) = 3

2r - 2rsin# = 3

Now make use of the identities y = rsin#, r^2 = x^2 + y^2

r=12/6-2sin(theta-pie/3)

Multiply through by (6 - 2sin(#-{pi/3})): r(6 - 2sin(#-{pi/3})) = 12

6r - 2rsin(# - pi/3) = 12

Expand sin(# - pi/3) using the formula for sin(A - B), a compound angle formula.

Also, use the identity y = rsin#, r^2 = x^2 + y^2

 

[vwguy]

One more thing, how do I put it in the form, Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
Thanks

 

[morson]

I've already given you the necessary information. Here's the 2nd one:

2r - 2rsin# = 3

2 * sqrt(x^2 + y^2) = 3 + 2y

Squaring:

4(x^2 + y^2) = 9 + 12y + 4y^2

4x^2 = 9 + 12y

==> 4x^2 - 12y - 9 = 0

Which is in that form. You should be able to do the rest.