Introduction to Locus

Analytical geometry lets us study geometric shapes — lines, circles, curves — using algebra and coordinates instead of a compass and ruler alone. Tamil Nadu's Business Mathematics syllabus covers the same coordinate-geometry principles taught nationally, applied here to problems a commerce student will meet again in cost/break-even graphs and in later statistics work.

A locus is the path traced out by a point that moves according to a fixed geometric rule. Instead of plotting the moving point one position at a time, we translate the rule into an algebraic equation in $x$ and $y$ — every point $(x, y)$ satisfying that equation lies on the locus, and no point outside it does.

The general method for finding the equation of a locus has four steps:

1. Let the moving point be $P(x, y)$.
2. Write the given geometric condition in words as a distance (or other measurable) relationship involving $P$.
3. Translate that relationship into an algebraic equation using the distance formula $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
4. Simplify the equation to its standard algebraic form.

Because the distance formula involves a square root, we usually square both sides early to remove it — this is safe as long as both sides are non-negative, which distances always are.