The position vector of a particle is given as $\vec r\, = \,({t^2}\, - \,8t\, + \,12)\,\hat i\,\, + \,\,{t^2}\hat j$ The time after which velocity vector and acceleration vector becomes perpendicular to each other is equal to........$sec$

For any arbitrary motion in space, which of the following relations are true

$(a)$ $\left. v _{\text {average }}=(1 / 2) \text { (v }\left(t_{1}\right)+ v \left(t_{2}\right)\right)$

$(b)$ $v _{\text {average }}=\left[ r \left(t_{2}\right)- r \left(t_{1}\right)\right] /\left(t_{2}-t_{1}\right)$

$(c)$ $v (t)= v (0)+ a t$

$(d)$ $r (t)= r (0)+ v (0) t+(1 / 2)$ a $t^{2}$

$(e)$ $a _{\text {merage }}=\left[ v \left(t_{2}\right)- v \left(t_{1}\right)\right] /\left(t_{2}-t_{1}\right)$

(The 'average' stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

The figure shows the velocity $(v)$ of a particle plotted against time $(t)$

A particle is moving with velocity $\vec v = K(y\hat i + x\hat j)$ where $K$ is a constant. The general equation for its path is