Two particles P and Q located at the points P t , t 3 - 16 t - 3 , Q t + 1 , t 3 - 6 t - 6 are moving in a…

$P\left(t,t^3-16t-3\right),Q\left(t+1,t^3-6t-6\right)$

$\Rightarrow PQ=\sqrt{(t+1-t{)}^2+{\left(t^3-6t-6-t^3+16t+3\right)}^2}$

$= \sqrt{1+(10t-3{)}^2}$

Let, $y=P{Q}^2=1+{\left(10t-3\right)}^2$

Minimum value of $y$ will be at $\frac{dy}{dt}=0$

i.e. $2\left(10t-3\right)=0$

or $t=\frac{3}{10}$

$\therefore$ minimum distance $= \sqrt{1+{\left(10\times \frac{3}{10}-3\right)}^2}=1$