# 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 $b=-4$.
- $x=-\frac{-4}{2\times 2}=1$
- Substitute $x=1$ into the equation to find $y$:
- $y=2\times {1}^{2}-4\times 1+3=2-4+3=1$
**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 $y=2{x}^{2}-4x+3$ 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.