Let $A$ and $B$ be the points on the coordinate axes, where the line intersect.


Since, $P$ divides the line segment $B A$ in the ratio $2: 3$.
Then, the coordinates of point $P$ will be
$$
\left(\frac{2 a+0}{2+3}, \frac{0+3 b}{2+3}\right) \text { i.e. }\left(\frac{2 a}{5}, \frac{3 b}{5}\right) .
$$
But coordinates of $P$ are $\left(\frac{1}{2}, \frac{1}{3}\right)$.
$$
\begin{aligned}
& \therefore \frac{2 a}{5}=\frac{1}{2} \text { and } \frac{3 b}{5}=\frac{1}{3} \Rightarrow a=\frac{5}{4} \\
& \text { and } \\
& b=\frac{5}{9} \\
&
\end{aligned}
$$
So, equation of line will be
$$
\begin{gathered}
\frac{x}{\frac{5}{4}}+\frac{y}{\frac{5}{9}}=1 \\
4 x+9 y=5
\end{gathered}
$$