Conic Sections — Ellipse and Hyperbola

• The standard equations of ellipse and hyperbola and their key parameters.
• Relation b² = a²(1−e²) for ellipse and b² = a²(e²−1) for hyperbola.
• Eccentricity as the defining property: e < 1 (ellipse), e > 1 (hyperbola), e = 1 (parabola).
• Foci, directrices, latus rectum, and asymptotes for each conic.

Level 1 — Ellipse standard form

Equation: x²/a² + y²/b² = 1 (a > b > 0 for horizontal major axis).

Key elements of ellipse (a > b):

Focal property: For any point P on ellipse, PF₁ + PF₂ = 2a (sum of focal distances constant).

Level 2 — Hyperbola standard form

Equation: x²/a² − y²/b² = 1 (horizontal transverse axis).

Key elements of hyperbola:

Focal property of hyperbola: |PF₁ − PF₂| = 2a (difference of focal distances = constant = 2a).

Rectangular hyperbola: a = b → e = √2. Asymptotes perpendicular. Equation xy = c² after rotation.

JEE tip: The crucial difference — ellipse: b² = a²−c² (b² < a²); hyperbola: b² = c²−a² (no restriction b < a).

NCERT spotlight — Conjugate hyperbola

Conjugate hyperbola: −x²/a² + y²/b² = 1 (same asymptotes y = ±(b/a)x as original). If e₁ and e₂ are eccentricities of a hyperbola and its conjugate: 1/e₁² + 1/e₂² = 1.

Parametric forms: Ellipse: (a cos θ, b sin θ). Hyperbola: (a sec θ, b tan θ) or (a cosh t, b sinh t).

Tangent at (x₁,y₁) on ellipse: xx₁/a² + yy₁/b² = 1. On hyperbola: xx₁/a² − yy₁/b² = 1.

Find the equation of the ellipse with foci at (±3, 0) and semi-major axis a = 5. Also find eccentricity, b, and length of latus rectum.

Step 1 — Given: c = 3, a = 5.
Step 2 — b² = a² − c² = 25 − 9 = 16 → b = 4.
Step 3 — Equation: x²/25 + y²/16 = 1.
Step 4 — Eccentricity: e = c/a = 3/5 = 0.6 < 1 ✓ (ellipse).
Step 5 — Latus rectum: 2b²/a = 2×16/5 = 32/5 = 6.4.
Step 6 — Check focal property: PF₁ + PF₂ = 2a = 10 for any point P on ellipse.

Applications — planetary orbits and GPS

Planetary orbits are ellipses with the Sun at one focus (Kepler's 1st Law). Earth's orbital eccentricity ≈ 0.017 (nearly circular). Whispering galleries use elliptical rooms — sound from one focus reaches the other. Hyperbolic navigation (LORAN): difference in time signals from two stations (foci) is constant along a hyperbola — determines position.

• Write the ellipse with semi-axes 6 and 4 (major axis along y).
• Find eccentricity and foci of x²/9 − y²/16 = 1.
• Prove that asymptotes of x²/a² − y²/b² = 1 are y = ±(b/a)x by setting b² = a²e²−a².

Open the Practice tab for graded questions on Ellipse and Hyperbola.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.