Example 3. Determine whether the point $\mathrm{P}(1, \pi)$ lies on the hyperbola
$$
\rho=\frac{2}{1-3 \cos \theta} \text { . }
$$
The general expression for the coordinates of point $\mathrm{P}$ is
$$
(1, \pi+2 \mathrm{k} \pi)
$$

or $(-1,2 \mathrm{k} \pi)$.
Substituting (1) into the curve equation, we get
$$
1=\frac{2}{1-3(-1)}=\frac{1}{2},
$$

which is a contradiction, so $\mathrm{k}$ does not exist.
Substituting (2) into the curve equation, we get
$$
-1=\frac{2}{1-3 \cdot 1}=-1 \text {, }
$$

which satisfies the curve equation for any integer $\mathrm{k}$, so point $\mathrm{P}$ lies on the curve.