Parabola

1. Definition of a Parabola:

  • A parabola is a U-shaped curve formed by intersecting a cone with a plane parallel to its side.
  • It is defined as the set of points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

2. Equation of a Parabola:

  • General equation: y=ax2+bx+c or x=ay2+by+c, where a, b, and c are constants..
  • Standard form for a vertical parabola with its vertex at (h,k): y=a(xh)2+k.
  • Standard form for a vertical axis of symmetry: y=ax2+bx+c.
  • Standard form for a horizontal axis of symmetry: x=ay2+by+c.
  • For a horizontal parabola, the equation becomes x=ay2+by+c or x=a(yk)2+h.

3. Elements of a Parabola:

  • Focus and Directrix:
    • Focus: A fixed point on the axis of symmetry, equidistant from any point on the parabola.
    • Directrix: A fixed line perpendicular to the axis of symmetry, also equidistant from any point on the parabola.
  • Axis of Symmetry:
    • A line that divides the parabola into two symmetric halves. For a vertical parabola, the axis is parallel to the y-axis; for a horizontal parabola, it's parallel to the x-axis.
  • Vertex:
    • The point where the parabola intersects its axis of symmetry. It's the minimum (for a downward-opening parabola) or maximum (for an upward-opening parabola) point on the graph.

4. Properties of Parabolas:

  • The parabola opens upward if the coefficient a is positive and downward if a is negative in the equation y=ax2+bx+c.
  • If the coefficient of the squared term is 1 (i.e., y=x2), the parabola is symmetric around the y-axis.
  • If the squared term is x2 in x=ay2+by+c, the parabola is symmetric around the x-axis.

5. Types of Parabolas:

  • Vertical Parabola:
    • Opens either upward (if a>0) or downward (if a<0).
    • Standard form: y=ax2+bx+c.
  • Horizontal Parabola:
    • Opens either rightward (if a>0) or leftward (if a<0).
    • Standard form: x=ay2+by+c.

 Applications of Parabolas:

  • Real-life applications include projectile motion (like the path of a thrown object), satellite dish designs, and reflecting telescopes.
  • Used in engineering for designing lenses and mirrors to focus light and in architecture for designing arches.