"Show that the polar equation $r = a \sin(t) + b \cos(t)$, where $ab \neq 0$, represents a circle, and find its center and radius."
I don't need the answer – I just need a hint to nudge me on. I've tried to tackle it by investigating tangents, completing squares, force-fitting into the formula of a circle, etc., but I can't seem to arrive at the indisputable conclusion that it must be a circle.
This is not homework, by the way. It comes from a Calculus textbook that I am studying on my own. I understand you have no reason to believe me, but if I were taking a real course, I'd bring this to my TA.
Best Answer
Multiply either side by $r$, use $r^2=x^2+y^2, r\sin t=y,r\cos t=x$
But we don't need $ab\ne 0$
If $a=0,r=b\cos t\implies r^2=b r\cos t\implies x^2+y^2-bx=0$ $\implies (x-\frac b 2)^2+y^2=(\frac b 2)^2$ is a circle.
If both $a,b$ are $0, r=0\implies x^2+y^2=0$ which is point-circle.