Question

Determine if the following trajectory lies on a circle...

I start with a vector I am down to 2(cos^2)t+4(sin^2)t+2sintcost-2sqrt(3)sintcost but am just bad enough at trig to not know what to do from here to find what the radius is this trajectory lies on...

Answer 1

Your algebra is wrong; if you square the formulas for x and y separately and add them, the sintcost terms will cancel, and you'll be left with 4(sin^2 t + cos^2 t), so the magnitude of your vector is constant for all values of t

Answer 2

you have a vector v= <x,y> where both x and y are functions of t
the length of the vector is
\[ \sqrt{v \cdot v} = \sqrt{x^2+y^2} \]
if this length is constant then we know all the points traced out by v over time must lie on the circumference of a circle.