The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

The condition for a line $y=mx+c$ to be a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:
$$c^2=a^2{m}^2-b^2$$
For the line $y=\alpha x+\beta$, we have $m=\alpha$ and $c=\beta$.Substituting these values:
$${\beta }^2=a^2{\alpha }^2-b^2$$
To find the locus of $(\alpha ,\beta )$, replace $\alpha$ with $x$ and $\beta$ with $y$:
$$y^2=a^2{x}^2-b^2$$
$$a^2{x}^2-y^2=b^2⟹\frac{x^2}{(b/a{)}^2}-\frac{y^2}{b^2}=1$$
This is the equation of a hyperbola.