If the radii of the circles  ${\left(x-1\right)}^2+{\left(y-2\right)}^2=1$ and  ${(x-7)}^2+{(y-10)}^2=4$ are increasing uniformly w.r.to time as 0.3 units/s and 0.4 units/s respectively, then they will touch each other at t =     

$S_1:{(x-1)}^2+{(y-2)}^2=1$

Centre : (1,2) , radius $\left(r_1\right)=1$

$S_2: {(x-7)}^2+{(y-10)}^2=4$

Centre (7,10), radius  $\left(r_2\right)=2$
 $\because c_1{c}_2=10>r_1+r_2$ 
 Hence, two circles are separated
 Radii of two circles at time ‘t’ are  $(1+0.3t)$ and  $(2+0.4t)$  
 For the two circles touch each other, then 
$$\begin{gathered} c_1{c}_2=|(1+0.3t)\pm (2+0.4t)| \\ \Rightarrow 10=|3+0.7t| or 10=|-1-0.1t| \\ \Rightarrow 0.7t+3=\pm 10 or -1-0.1t=\pm 10 \\ \Rightarrow t=10 or t=50 \end{gathered}$$