HSSLIVE Plus One Maths Chapter 11: Conic Sections Notes

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Conic sections—circles, ellipses, parabolas, and hyperbolas—are curves formed by intersecting a cone with a plane. This chapter explores the geometric properties and algebraic representations of these curves, their standard forms, and transformations. Students will learn to identify and graph conics from their equations, understand their focal properties, and apply these concepts to real-world scenarios such as planetary orbits, optical systems, and architectural design. This knowledge connects algebra, geometry, and physical applications in a unified framework.

Chapter 11: Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane.

Circle:

A circle is the locus of points equidistant from a fixed point (center).

  • Standard Equation: (x – h)² + (y – k)² = r²
    • (h, k) is the center
    • r is the radius
  • General Form: x² + y² + 2gx + 2fy + c = 0
    • Center: (-g, -f)
    • Radius: √(g² + f² – c)

Parabola:

A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix).

  • Standard Equations:
    • When vertex at origin:
      • y² = 4ax (axis along positive x-axis)
      • x² = 4ay (axis along positive y-axis)
      • y² = -4ax (axis along negative x-axis)
      • x² = -4ay (axis along negative y-axis)
    • Focus: (a, 0) or (0, a)
    • Directrix: x = -a or y = -a

Ellipse:

An ellipse is the locus of points where the sum of distances from two fixed points (foci) is constant.

  • Standard Equation: x²/a² + y²/b² = 1 (a > b)
    • Center: (0, 0)
    • Major axis: 2a (along x-axis)
    • Minor axis: 2b (along y-axis)
    • Foci: (±c, 0) where c² = a² – b²
    • Eccentricity: e = c/a

Hyperbola:

A hyperbola is the locus of points where the difference of distances from two fixed points (foci) is constant.

  • Standard Equation: x²/a² – y²/b² = 1
    • Center: (0, 0)
    • Transverse axis: 2a (along x-axis)
    • Conjugate axis: 2b (along y-axis)
    • Foci: (±c, 0) where c² = a² + b²
    • Asymptotes: y = ±(b/a)x
    • Eccentricity: e = c/a

 

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