Example 3. Find the locus of the intersection of the two tangents to the hyperbola $\frac{x^{2}}{2}-y^{2}$ $=1$ at the points of intersection with the line passing through (1) $\mathrm{P}_{1}^{\prime}\left(2, \frac{1}{2}\right)$, (2) $P_{2}^{\prime}(-2,3)$.
(1) Let $\mathrm{A}=\frac{1}{2}, \mathrm{C}=-1, \mathrm{D}=0$,
$$
\mathrm{E}=0, \mathrm{~F}=-1, \mathrm{x}^{\prime}=2, \mathrm{y}^{\prime}=\frac{1}{2} \text { into }
$$

we get $x-\frac{1}{2} y-1=$
0, which simplifies to $2 x-y-2$
$$
=0\left(1_{1}\right) \text {. }
$$

Since point $P_{1}$ is
inside the
hyperbola, the
required trajectory is the line 11 (Figure 4).
$$
\begin{array}{l}
\text { (2) Let } \mathrm{A}=\frac{1}{2}, \\
\mathrm{C}=-1, \mathrm{D}=0, \\
\mathrm{E}=0, \mathrm{~F}=-1, \mathrm{x}^{\prime}=-2, \mathrm{y}^{\prime}=3 \text { into }
\end{array}
$$
(i), we get $x+3 y+1=0 \quad\left(l_{2}\right)$.

Since $\mathrm{P}_{2}$ is outside the hyperbola, the required trajectory is the segment $\mathrm{MN}$ of line $\mathrm{I}_{2}$ outside the hyperbola (Figure 4).