JEE Mathematics: Complex Numbers and Coordinate Geometry
Complex Numbers and Coordinate Geometry together constitute approximately 25–30% of JEE Advanced Mathematics marks. Complex number operations: for z = a + ib, the modulus |z| = √(a²+b²) and argument θ = arctan(b/a) — careful about the quadrant. De Moivre's Theorem: (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ) — used to evaluate zⁿ for large n and to find nth roots of complex numbers. The nth roots of unity are z_k = e^(2πik/n) = cos(2πk/n) + i sin(2πk/n) for k = 0, 1, ..., n−1 — they form a regular n-gon on the unit circle. Coordinate Geometry advanced topics: locus problems (finding the equation of the curve traced by a point satisfying a given condition), properties of conics (tangent and normal to ellipse, parabola, hyperbola), and the parametric form of curves. For parabola y² = 4ax: parametric point (at², 2at); tangent at parameter t: ty = x + at²; normal: y = −tx + 2at + at³. JEE frequently tests the 'foot of perpendicular from a point to a tangent' or 'intersection of normals' — draw a diagram first and identify the geometric relationship before algebraic manipulation.