$\text{ The eccentricity of the locus of point }(3h+2,k)\text{ , where }(h,k)\text{ lies on the circle }{x}^2+y^2=1,\text{ is }$

$\begin{array}{l}\text{ Let }p=3h+2\text{ and }q=k\\ \text{ or }h=\frac{p-2}{3}\text{ and }k=q\end{array}$

$\text{ Since }(h,k)\text{ lies on }{x}^2+y^2=1\text{ , we have }{h}^2+k^2=1$

$\text{ or }{\left(\frac{p-2}{3}\right)}^2+q^2=1$

$\text{The locus is}{\left(\frac{x-2}{3}\right)}^2+y^2=1$

$\text{which has eccentricity }e=\sqrt{1-\frac{1}{9}}=\frac{2\sqrt{2}}{3}$