Let, $A=\left(a,0\right)$ and $B=\left(0,b\right)$, then the equation of $AB$ is $\frac{x}{a}+\frac{y}{b}=1$ which passes through $\left(\frac{6}{5},\frac{6}{5}\right)$

$\Rightarrow \frac{1}{a}+\frac{1}{b}=\frac{5}{6}$
Let, $P=\left(h,k\right)$, then

$h=\frac{2\left(0\right)+1\left(a\right)}{3}=\frac{a}{3}$ & $k=\frac{2\left(b\right)+1\left(0\right)}{3}=\frac{2b}{3}$
$\Rightarrow a=3h$ & $b=3\mathrm{}\frac{k}{2}$
$\Rightarrow \frac{1}{3h}+\frac{2}{3k}=\frac{5}{6}$
$\Rightarrow$ locus of $P$ is $2\left(y+2x\right)=5xy$