Locus ProblemsJEE Main

Conditions become curves — interactive Mathematics simulation for IIT-JEE.

Concept

A locus is the set of all points satisfying a geometric condition — conditions BECOME curves. Equidistance from two points gives a line; a fixed ratio of distances gives the Apollonius circle; point-vs-line equidistance a parabola; constant sum an ellipse. The universal recipe: call the moving point (h, k), write the condition in coordinates, simplify, rename to (x, y).

Key formula

P = (h, k) \to \text{condition in } h, k \to \text{algebra} \to \text{equation in } x, y

Derivation

Apollonius (PA = 2PB, A(−3,0), B(3,0)): $(h+3)^2 + k^2 = 4[(h-3)^2 + k^2]$ → $3h^2 + 3k^2 - 30h + 27 = 0$ → $(h-5)^2 + k^2 = 16$ — a circle, centre (5,0), radius 4.

Sliding ladder: endpoints (a, 0), (0, b) with a² + b² = 36; midpoint (a/2, b/2) ⇒ h² + k² = 9. The CONSTRAINT (ladder length) transfers to the tracked point.

Scenarios to explore

Locus Problems — Geometric conditions become curves — watch P trace them.

Real-world applications

Every conic definition IS a locus problem.
Robotics & mechanism design: coupler curves of linkages.
Radio direction finding (constant-ratio and constant-difference loci).

JEE exam tips

PA/PB = k: circle for k ≠ 1, line for k = 1 — the complete dichotomy.
Midpoint loci: express endpoint coordinates via the midpoint, feed the constraint.
Loci from moving tangents/chords: parametrise the moving object, then eliminate.

Common mistakes

Forgetting to eliminate the parameter (answers must involve x, y only).
Not squaring carefully — extraneous branches sneak in.
Dropping domain limits (the ladder midpoint traces only a QUARTER circle).

Exam traps to avoid

Locus of a point at constant distance from a LINE is a PAIR of parallel lines.
Right-angle subtension of a fixed segment gives a circle WITHOUT its two endpoints.