What are the parabolas and hyperbolas of Fermat?

Pierre de Fermat generalized the equations for 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 classification, the curves are referred to as parabolas or hyperbolas of Fermat depending on the exponent $n$:

• Parabolas of Fermat: These occur when $n$ is a positive value.1
• Hyperbolas of Fermat: These occur when $n$ is a negative value.1

Fermat’s study of these curves was significant because it led him to develop an algorithm equivalent to differentiation, which allowed him to determine tangents to curves and locate their maximum, minimum, and inflection points. Additionally, he utilized a summation process to find the areas bounded by these curves, a method equivalent to integral calculus.1