Ellipse & Hyperbola

Concept overview

An ellipse in standard form x²/a² + y²/b² = 1 (a > b) has foci (±c, 0) where c² = a² − b². Eccentricity e = c/a < 1. The sum of distances from any point on the ellipse to the two foci is 2a — the defining property.

A hyperbola x²/a² − y²/b² = 1 has foci (±c, 0) where c² = a² + b². Eccentricity e = c/a > 1. The difference of distances from any point to the two foci is 2a in absolute value. Asymptotes are y = ±(b/a) x.

For both, parametric form is (a cos θ, b sin θ) for ellipse and (a sec θ, b tan θ) for hyperbola. Tangent at (a cos θ, b sin θ) on ellipse: (x cos θ)/a + (y sin θ)/b = 1. Similar structure for hyperbola.

Latus rectum length = 2b²/a for both. Directrix at x = ±a/e. Focal-chord and reflection properties are standard JEE test material — for an ellipse, reflecting a ray from one focus off the ellipse sends it to the other focus.

Key formulas & concepts

• Ellipse standard: x²/a² + y²/b² = 1, a > b; foci (±c, 0); c² = a² − b²
• Hyperbola standard: x²/a² − y²/b² = 1; foci (±c, 0); c² = a² + b²
• Eccentricity: e = c/a; ellipse: e < 1; hyperbola: e > 1
• Ellipse parametric: (a cos θ, b sin θ)
• Hyperbola parametric: (a sec θ, b tan θ)
• Ellipse tangent at point: (x cos θ)/a + (y sin θ)/b = 1
• Hyperbola asymptotes: y = ±(b/a) x

Common mistakes to avoid

• Confusing c² = a² − b² (ellipse) with c² = a² + b² (hyperbola) — signs flip.
• Using tangent formulas at the wrong parametric angle.
• Assigning e < 1 to hyperbola or e > 1 to ellipse — reverse.
• Ignoring the axis orientation when b > a (major axis becomes vertical).

JEE Main relevance

JEE Main asks 1–2 ellipse/hyperbola problems every paper — usually one eccentricity or tangent problem and often one focal-chord or directrix question.

JEE Advanced relevance

Advanced problems combine multiple conics (an ellipse tangent to a circle) or use parametric-form clever substitutions for locus problems.

Previous-year questions (PYQ)

Eccentricity and tangent problems are the most repeated PYQ archetypes for both curves.