1. r = acos(x) + bsin(x)

I don't like the form of this equation, because it has cos(x) and sin(x), but we're trying to convert this polar equation into rectangular form (which contains x's and y's), so the angle needs to be something other than x. Let's use u instead.

r = acos(u) + bsin(u)

Now, I assume you know that the radius r can be considered the hypotonuse of a right triangle where the 2 legs are x and y and the angle is u. Knowing this, we can use a few identities to convert this equation:

cos(u) = x/r -> rcos(u) = x

sin(u) = y/r -> rsin(u) = y

x^2 + y^2 = r^2

Let's take the equation r = acos(u) + bsin(u), and multiply both sides by r:

r^2 = arcos(u) + brsin(u) ... look at what we have: r^2, rcos(u), rsin(u). We can substitute the above relations to get:

x^2 + y^2 = ax + by

x^2 - ax + y^2 - by = 0

Now by completing the square, we get:

[x^2 - ax + (a/2)^2] + [y^2 - by + (b/2)^2] = (a/2)^2 + (b/2)^2

(x - a/2)^2 + (y - b/2)^2 = (a^2 + b^2)/4

Center is (a/2,b/2)

Radius = sqrt(a^2 + b^2)/2