Chapter 10 — Conic Sections

Conic sections are curves obtained by intersecting a right circular cone with a plane. They include circles, ellipses, parabolas, and hyperbolas, each with distinct geometric properties and algebraic equations used in applications like planetary motion and reflector design.

Chapter 10 introduces conic sections as curves formed by the intersection of a plane with a double-napped right circular cone. The chapter covers four main types: circles (all points equidistant from a fixed center), parabolas (points equidistant from a focus and directrix), ellipses (sum of distances from two foci is constant), and hyperbolas (difference of distances from two foci is constant). Each curve has standard equations with vertex at the origin and axes along coordinate axes. Additional concepts include latus rectum (perpendicular chord through the focus), eccentricity (ratio describing curve shape), and degenerate cases (point, line, or pair of intersecting lines) when the cutting plane passes through the cone's vertex. Applications extend to telescope design, automotive headlights, suspension bridges, and outer space exploration.

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