Chapter 10 — Conic Sections

Chapter 10 of the Class 11 Maths NCERT textbook, "Conic Sections", covers curves formed by intersecting a right circular cone with a plane — circles, ellipses, parabolas, and hyperbolas — each with distinct geometric properties and equations used in planetary motion and reflector design.

• One cone, four curves — The chapter's unifying idea is that circles, parabolas, ellipses and hyperbolas all arise from slicing a double cone at different angles, so seemingly unrelated curves share a single geometric origin.
• Defining each conic by distance — Each curve is characterised by a distance condition — equidistant from a centre, from a focus and directrix, or by constant sum or difference of distances from two foci — giving a clean geometric definition behind every standard equation.
• Shape descriptors and degenerate cases — Concepts like eccentricity and latus rectum quantify a conic's shape, while degenerate cases (a point, line or crossed lines) appear when the plane passes through the vertex, rounding out when a conic collapses.

1. 01A circle: (x – h)² + (y – k)² = r², where (h, k) is the center and r is the radius
2. 02A parabola: y² = 4ax (opening rightward), where the focus is at (a, 0) and directrix is x = –a; latus rectum length = 4a
3. 03An ellipse: x²/a² + y²/b² = 1 (foci on x-axis), where c² = a² – b² and eccentricity e = c/a; latus rectum length = 2b²/a
4. 04A hyperbola: x²/a² – y²/b² = 1 (transverse axis on x-axis), where c² = a² + b² and eccentricity e = c/a (always > 1); latus rectum length = 2b²/a
5. 05Degenerate cases occur when the plane cuts through the cone's vertex: a point (α < β ≤ 90°), a straight line (β = α), or pair of intersecting lines (0 ≤ β < α)
6. 06Real-world applications: parabolic mirrors in flashlights and automobile headlights, suspension bridge cables, arches in bridges, and planetary orbits