How did Pierre de Fermat use algebra to study geometric loci?

Pierre de Fermat facilitated the study of geometric loci—sets of points with specific characteristics—by applying algebra to geometry through the use of a coordinate system.1

In his paper Ad Locos Planos et Solidos Isagoge (1636), Fermat considered an equation involving two variables. He represented one variable as a line measured horizontally from a fixed point, while the second variable was represented by a line positioned at the end of the first and inclined at a fixed angle. As the first variable changed, the second took on a value determined by the equation, causing the endpoint of the second line to trace a curve in space.2

This construction allowed Fermat to establish the fundamental principle of analytic geometry: whenever two unknown quantities are found in an equation, they define a fixed locus, with the endpoint of one quantity describing a straight line or a curve. Through this method, he demonstrated that quadratic equations correspond to conic sections, such as the ellipse, parabola, or hyperbola.2