Parabola Deep DiveJEE Main

Concept

A parabola is the set of points equidistant from a focus and a directrix. For y² = 4ax: focus (a, 0), directrix x = −a, and the golden parametrisation (at², 2at) that turns geometry into algebra. Focal distance is simply x + a — the definition, restated.

Key formula

Derivation

Definition: √((x−a)² + y²) = x + a. Square → y² = 4ax.

Parametric point satisfies it: (2at)² = 4a·at² ✓. Chord through the focus: the line joining t₁, t₂ passes (a, 0) ⟺ t₁t₂ = −1 (substitute into the chord equation). Tangent by implicit differentiation: 2yy′ = 4a → slope 1/t at the point.

Scenarios to explore

• Parabola Deep Dive — Focus, directrix, latus rectum and the t-parametrisation.

Real-world applications

• Satellite dishes & headlights: rays through the focus emerge parallel (reflection property).
• Projectile paths are parabolas.
• Suspension-bridge cables under uniform load.

JEE exam tips

• Focal chord length = a(t + 1/t)² — minimum 4a (the latus rectum).
• Tangents at t₁, t₂ meet at (at₁t₂, a(t₁+t₂)) — intersection formulas by symmetric functions.
• Point of tangency doubles: tangent from external point exists iff point lies outside (S₁ > 0).

Common mistakes

• Latus rectum = 4a, not 2a (it's the full chord through the focus).
• Sign conventions for left/up/down-opening variants (y² = −4ax, x² = ±4ay).
• Semi-latus rectum vs latus rectum in formula lookups.

Exam traps to avoid

• Tangents at the ends of a focal chord meet ON the directrix — at right angles!
• x² = 4ay opens UP with focus (0, a): don't recycle y²-form formulas blindly.