Geometric loci in the Cartesian plane

Many problems in coordinate geometry ask us to find the equation of a curve starting from a property that characterises its points: which are the points equidistant from the endpoints of a segment? Those equidistant from a point and a line? Those at a fixed distance from a centre? The answers — the perpendicular bisector of the segment, the parabola, the circle — are familiar curves, but in this chapter we derive them from their locus definition. The method is always the same: take a generic point $P(x;y)$, translate the property into an equality between distances, square the terms containing a root (allowed, since distances are $\ge 0$) and simplify down to an equation in $x$ and $y$. At the end we see how the parabola, the ellipse, the hyperbola and the circle are all instances of a single scheme founded on distances.

Sections

• Locus
• The bisector of an angle
• The perpendicular bisector of a segment
• The parabola as a locus
• A unifying definition of the conics

Exercises

• All the chapter’s exercises