Locus Problems in Coordinate Geometry for JEE Main

A locus is the set of all points satisfying a geometric condition, and finding its equation is a fundamental skill that bridges coordinate geometry with analysis. JEE Main tests locus in every session — sometimes as a standalone question (find the equation of the locus), sometimes embedded in a conic section or circle problem. The four-step method described here handles every type reliably.

The Four-Step Method

Step 1: Let the moving point be P(h, k). Step 2: Write the given geometric condition algebraically in terms of h, k, and the fixed elements. Step 3: Simplify, eliminating any parameter if present. Step 4: Replace h with x and k with y to write the locus equation. This procedure never changes — only the condition in step 2 varies. Working from a moving point (h, k) is a discipline that prevents the most common setup errors. For the coordinate geometry foundations see our coordinate geometry guide.

Distance-Based Loci

Equidistant from two fixed points: locus is the perpendicular bisector (a line). Fixed distance from a fixed point: locus is a circle. Distances to two fixed points in a fixed ratio k : 1 (k ≠ 1): locus is the Apollonius circle. Distances to two fixed points summing to a constant 2a (> distance between them): ellipse with those points as foci. Difference of distances = constant 2a: hyperbola. These five standard distance loci should be identifiable at a glance from the condition statement.

Parametric Loci

When the point's coordinates are given parametrically, eliminate the parameter between x(t) and y(t). For a point P = (at², 2at) on a parabola y² = 4ax, eliminating t: from y = 2at get t = y/(2a), substitute x = at² = a(y/(2a))² = y²/(4a), recovering y² = 4ax. Parametric loci arise in problems about points on a curve whose tangent satisfies some condition, or about the midpoint of a chord.

Midpoint of Chords and Chord of Contact

For a chord of a circle joining two points (x₁, y₁) and (x₂, y₂), the midpoint (h, k) satisfies a specific equation involving the chord's slope equalling the negative ratio of gradients — the T = S₁ concept. Similarly, the chord of contact from an external point (x₁, y₁) to a conic is given by replacing the conic equation's x², y² with x₁x, y₁y and the linear terms. These chord locus results are direct JEE question patterns.

Intersection of Variable Lines

A family of lines parameterised by t intersects each other at moving points — finding their locus requires eliminating t between the x and y coordinates of the intersection. For example, a line through fixed point A and rotating, intersecting a fixed line — the locus of the intersection traces a line (or conic) depending on the constraint. These problems test the combined skill of finding intersections and eliminating parameters, which is the highest-difficulty locus type. After practising all four locus categories, take a free mock test on coordinate geometry.