Worked Loci: Distance, Ratio, Angle and Area Conditions

This section applies the five-step method to the four condition-types that recur throughout the exercise: equal distance from two fixed points, a fixed ratio of two distances, a right angle subtended at the moving point, and a constant area for a triangle formed with two fixed points.

Equal-distance loci. If $P$ is equidistant from two fixed points $A$ and $B$, Step 3 gives $PA^2 = PB^2$; squaring the distance formula on both sides and simplifying always cancels the $x^2$ and $y^2$ terms, leaving a linear equation -- the locus is the perpendicular bisector of $AB$. This is the simplest case and is a good first check that the method is being applied correctly: if a locus problem states an equal-distance condition and the simplified equation is not linear, an algebra slip has occurred.

Ratio (Apollonius) loci. If $PA:PB = m:n$ with $m \neq n$, Step 3 gives $n^2 PA^2 = m^2 PB^2$; simplifying no longer cancels the quadratic terms (since the two sides carry different constant multiples), and the locus is a circle, called an Apollonius circle, whose centre lies on the line $AB$ produced. When $m=n$ the same algebra degenerates to the equal-distance case above.

Right-angle (Thales) loci. If a fixed segment $AB$ subtends a right angle at the moving point $P$, the condition (slope of $PA$)$\times$(slope of $PB$)$=-1$, or equivalently $PA^2+PB^2=AB^2$, always simplifies to a circle with $AB$ as diameter -- this is the coordinate-geometry form of Thales' theorem, and it is worth recognising on sight: whenever a segment subtends a right angle at a variable point, the answer is "the circle on that segment as diameter," and the algebra should confirm it. …