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 $y=a{x}^{2}+bx+c$, 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 $y=2{x}^{2}-4x+3$.

1. Finding the Vertex:

• The equation is in the form $y=a{x}^{2}+bx+c$, where the vertex is given by $x=-\frac{b}{2a}$.
• $a=2$ and $b=-4$.
• $x=-\frac{-4}{2×2}=1$
• Substitute $x=1$ into the equation to find $y$:
• $y=2×{1}^{2}-4×1+3=2-4+3=1$
• Vertex: $\left(1,1\right)$

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 $\left(1,1\right)$, the distance between its endpoints can be found by considering points on the parabola.

4. Graphical Representation:

• Plot the parabola $y=2{x}^{2}-4x+3$ on a graph.
• Identify the vertex at $\left(1,1\right)$ 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 $\left(0,1\right)$ 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.