Parabola

Concept overview

A parabola is the locus of points equidistant from a fixed focus and a fixed directrix line. In standard orientation y² = 4ax, the focus is (a, 0), directrix is x = −a, vertex is (0, 0), axis is the x-axis, and the latus rectum (chord through focus perpendicular to axis) has length 4a.

Parametric form: (at², 2at). Tangent at (at², 2at) is ty = x + at². Normal at this point is y + tx = 2at + at³.

Chord of contact from external point (x₁, y₁): yy₁ = 2a(x + x₁). The line y = mx + c is tangent iff c = a/m (tangent condition).

Reflection property: a ray parallel to the axis reflects off the parabola and passes through the focus — the design principle of parabolic mirrors and dish antennas.

Key formulas & concepts

• Standard equation: y² = 4ax; focus (a, 0); directrix x = −a
• Parametric form: (at², 2at)
• Tangent at t: ty = x + at²
• Normal at t: y + tx = 2at + at³
• Tangent condition (y = mx + c): c = a/m
• Latus rectum: = 4a; ends at (a, ±2a)

Common mistakes to avoid

• Using tangent equation for wrong form — y² = 4ax and x² = 4ay have different tangent formulas.
• Forgetting the axis of symmetry when the parabola opens sideways vs upward.
• Confusing directrix with axis — they're perpendicular and play different roles.
• Applying focal chord properties without verifying the chord passes through the focus.

JEE Main relevance

JEE Main asks 1 parabola problem every paper — usually a tangent/normal or a focal chord property.

JEE Advanced relevance

Advanced problems combine parabolas with other conics or ask about locus problems involving parabola-tangent triangles.

Previous-year questions (PYQ)

Tangent and normal at a parametric point are the top-recurring PYQ archetypes.