Cartesian equation, focus and directrices

Cartesian Equation

A Cartesian equation is an equation that identifies each point uniquely in a space by an ordered set of coordinates related through mathematical operations.
For instance, in a two-dimensional plane, a point can be indicated as (x,y), where x signifies the distance along the x-coordinate and y is the distance along the y-coordinate.
These coordinates represent an intersection of horizontal and vertical lines in a rectangular grid.
The Cartesian equation of a line in a two-dimensional plane is represented in the form y = mx + c, where m is the slope of the line and c is the y-intercept.

Focus

The focus in the context of conic sections refers to a particular point(s) at which rays of light meet or from which they appear to diverge after being reflected by a curve.
For instance, in a parabola, there exists one focus. The set of all points equidistant from this focus and a line termed the directrix forms the parabola.
Ellipses and hyperbolas, however, have two foci. In an ellipse, the total distance from the two foci to any point on the ellipse remains constant. Conversely, in a hyperbola, the absolute difference of the distances from the two foci to any point on the hyperbola remains constant.

Directrix

A directrix, in relation to conic sections, is a reference line from which distances are calculated in generating a conic.
This takes into account the principle that a conic section represents the locus of a point that moves so that its distance from a pre-defined point, the focus, is in a constant ratio to its distance from the directrix.
For example, in the case of a parabola, each point on the curve is equally far from the directrix as it is from the focus.
With regard to ellipses and hyperbolas, each focus is associated with a distinct directrix, and the constant ratio of distances for each point on the curve corresponds to the distance from the point to the nearer directrix divided by the distance to the nearer focus.

Understanding the above concepts is essential as they provide the foundation for exploring more complex topics within Further Pure Mathematics.