How did Pierre de Fermat generalize the equations of a parabola and a hyperbola?

Pierre de Fermat generalized the equations of the ordinary parabola ($ay=x^2$) and the rectangular hyperbola ($xy=a^2$) into the unified form:

$$a^{n-1}y=x^n$$

In this generalization, the resulting curves are classified as parabolas or hyperbolas of Fermat depending on whether the exponent $n$ is positive or negative, respectively. This work was part of Fermat's broader development of analytic geometry, where he emphasized that any algebraic relation between $x$ and $y$ coordinates defines a curve. By applying these algebraic techniques, Fermat was able to move beyond the specific geometric constraints of classical antiquity and analyze a much wider class of curves.123