Particle moving on a circle: position and velocity vectors

Write $5 - 6\sin^2(t)$ in the form $\alpha + \beta\cos(2t)$, where $\alpha, \beta \in Z^+$.

Show that the Cartesian equation of the path of the particle is $(x-2)^2 + (y-1)^2 = 9$.

The particle is at point $A$ when $t = 0$ and at point $B$ when $t = a$, where $a$ is a positive real constant.

If the distance travelled along the curve from $A$ to $B$ is $\frac{3\pi}{4}$, find $a$.

Find all values of $t$ for which the position vector of the particle, $\tvec{r}(t)$, is perpendicular to its velocity vector, $\dot{\tvec{r}}(t)$.