Determine the standard equation of the parabola given its vertex(5, 6.5) and directrix(7.5) then determin its focus and axis of symmetry

To find the standard equation of the parabola given the vertex and directrix, we can follow these steps:

### 1. **Identify the given information:**
   - Vertex: \( (5, 6.5) \)
   - Directrix: \( x = 7.5 \)

### 2. **Determine the orientation:**
   Since the directrix is a vertical line (\(x = 7.5\)), the parabola opens either to the left or right. 

### 3. **Find the equation of the parabola:**
   The standard form for a parabola that opens horizontally is:

   \[
   (y - k)^2 = 4p(x - h)
   \]

   Where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus (or to the directrix).

### 4. **Calculate the value of \( p \):**
   - The distance between the vertex and the directrix is \( |h - 7.5| = |5 - 7.5| = 2.5 \).
   - Since the directrix is to the right of the vertex, \( p = -2.5 \).

### 5. **Substitute the known values into the equation:**
   \[
   (y - 6.5)^2 = 4(-2.5)(x - 5)
   \]
   Simplifying the equation:
   \[
   (y - 6.5)^2 = -10(x - 5)
   \]

### 6. **Find the focus:**
   The focus is \( p \) units to the left of the vertex since \( p \) is negative. Therefore, the focus is:
   \[
   (h + p, k) = (5 - 2.5, 6.5) = (2.5, 6.5)
   \]

### 7. **Axis of symmetry:**
   The axis of symmetry is the line passing through the vertex and the focus, which in this case is horizontal, so:
   \[
   y = 6.5
   \]

### Summary:
- **Standard equation of the parabola:** \((y - 6.5)^2 = -10(x - 5)\)
- **Focus:** \( (2.5, 6.5) \)
- **Axis of symmetry:** \( y = 6.5 \)

Would you like any further details or explanations? 

Here are five related questions:
1. How does the value of \( p \) affect the direction in which a parabola opens?
2. What is the geometric significance of the focus in a parabola?
3. Can you derive the equation for a vertically oriented parabola?
4. How would the equation change if the directrix was to the left of the vertex?
5. What would the parabola's equation look like if the vertex were at the origin?

**Tip:** The distance between the vertex and directrix determines the "width" of the parabola; the larger the distance, the wider the parabola opens.

Learn how to find the standard equation of a parabola given its vertex at (5, 6.5) and directrix at x = 7.5. This problem involves determining the focus, axis of symmetry, and understanding the relationship between the vertex, directrix, and focus of a parabola. Suitable for high school students studying conic sections.

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