The Five-Step Method for Finding a Locus

Every locus problem in this chapter is solved by the same five-step procedure, and learning to apply it mechanically removes almost all of the difficulty from the topic.

Step 1 -- form the point. Let $P(x,y)$ be an arbitrary point on the locus; $x,y$ are the coordinates of a general, moving point, not a fixed one.

Step 2 -- write the condition. State the geometric rule $P$ must obey -- equal distance from two given points, a stated ratio of two distances, a right angle subtended by a fixed segment, a fixed triangle area, and so on.

Step 3 -- translate into algebra. Replace each geometric quantity by its coordinate-geometry formula: a distance $PA$ becomes $\sqrt{(x-x_1)^2+(y-y_1)^2}$, a right angle becomes (slope of $PA$)$\times$(slope of $PB$)$=-1$, and an area becomes the standard determinant formula for a triangle's area.

Step 4 -- simplify. Clear radicals (usually by squaring both sides), remove brackets, collect like terms, and reduce to a standard linear or quadratic (circle) form.

Step 5 -- verify the converse. Confirm the simplified equation is fully equivalent to the original condition -- that squaring has not smuggled in extra points, and no genuine point has been lost. …