Diameter of a Parabola

Diameter of a Parabola:

1. Definition:

  • The diameter of a parabola is the longest chord passing through the vertex of the parabola.
  • It's a special case of a chord where the chord's endpoints coincide with the vertex.

2. Properties of the Diameter:

  • Passes Through Vertex: The diameter passes through the vertex of the parabola.
  • Length: It's the longest chord among all chords of the parabola.
  • Symmetry: The diameter is symmetric about the axis of the parabola.

3. Equation of the Diameter:

  • For a parabola y=ax2+bx+c, the equation of the diameter passing through the vertex is y=k, where k is the y-coordinate of the vertex.
  • For a horizontal parabola, the equation of the diameter passing through the vertex is x=h, where h is the x-coordinate of the vertex.

4. Finding the Length of the Diameter:

  • The distance between the endpoints of the diameter passing through the vertex is the length of the diameter.
  • Calculate the distance using the coordinates of the vertex and the points where the diameter intersects the parabola.

5. Graphical Representation:

  • Plot the parabola on a graph.
  • Identify the vertex and visualize the line passing through the vertex, which represents the diameter.

Example:

Consider the parabola y=2x24x+3.

1. Finding the Vertex:

  • The equation is in the form y=ax2+bx+c, where the vertex is given by x=b2a.
  • a=2 and b=4.
  • x=42×2=1
  • Substitute x=1 into the equation to find y:
  • y=2×124×1+3=24+3=1
  • Vertex: (1,1)

2. Equation of the Diameter:

  • For this parabola, the equation of the diameter passing through the vertex is y=1.
  • This line is a special chord, the longest chord passing through the vertex.

3. Length of the Diameter:

  • The length of the diameter can be calculated as the distance between the endpoints.
  • Given that the diameter passes through the vertex at (1,1), the distance between its endpoints can be found by considering points on the parabola.

4. Graphical Representation:

  • Plot the parabola y=2x24x+3 on a graph.
  • Identify the vertex at (1,1) and visualize the line y=1 passing through it, representing the diameter.

Analysis:

  • The equation of the diameter passing through the vertex is y=1, indicating that the line passes through the point (0,1) on the y-axis.
  • This diameter is the longest chord of the parabola and is symmetric about the parabola's axis.
  • The length of the diameter can be calculated by finding the distance between the endpoints where the diameter intersects the parabola.