Length of latus rectum and the end points 

The length of the latus rectum and its endpoints can be determined from the standard form of the parabola.

### Step 1: Length of the Latus Rectum
For a parabola in the standard form \((y - k)^2 = 4p(x - h)\), the length of the latus rectum is given by \( |4p| \).

Given the equation:
\[
(y - 4)^2 = 8(x - 3)
\]

Here, \(4p = 8\), so the length of the latus rectum is:
\[
|4p| = 8
\]

### Step 2: Finding the Endpoints of the Latus Rectum
The latus rectum is a line segment perpendicular to the axis of symmetry and passes through the focus. The axis of symmetry for this parabola is horizontal (parallel to the x-axis), meaning the latus rectum is a vertical line.

#### 1. **Find the Focus:**
   - The focus of the parabola is at \((h + p, k)\), where \((h, k)\) is the vertex.
   - The vertex is \((3, 4)\) and \(p = 2\), so the focus is at \((3 + 2, 4) = (5, 4)\).

#### 2. **Endpoints of the Latus Rectum:**
   - The endpoints of the latus rectum will have the same x-coordinate as the focus, but the y-coordinates will be \(k + 2p\) and \(k - 2p\).
   - So the endpoints are:
     \[
     \left(5, 4 + 2\right) = (5, 6)
     \]
     \[
     \left(5, 4 - 2\right) = (5, 2)
     \]

### Summary:
- **Length of the Latus Rectum**: \(8\) units
- **Endpoints of the Latus Rectum**: \((5, 6)\) and \((5, 2)\)

Would you like further details or have any questions on this? Here are some related questions you might consider:

1. What are the properties of the latus rectum for a parabola that opens vertically?
2. How do the endpoints of the latus rectum change if the parabola is shifted?
3. Can the latus rectum ever lie entirely on one side of the vertex?
4. How would the endpoints change if \(p\) were negative?
5. What is the relationship between the latus rectum and the directrix?

**Tip:** The length of the latus rectum is directly related to the parameter \(p\), which indicates how "wide" or "narrow" the parabola is.

Learn how to find the length of the latus rectum and the endpoints of a parabola using its standard form equation. This problem involves understanding parabolas, their vertices, focuses, and how to calculate latus rectum lengths. Suitable for high school students studying conic sections.

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