Give the focus of the parabola given the vertex(5, 6.5) and a directtrix of 7.5

To find the focus of a parabola given its vertex and directrix, we need to use the property that the distance from any point on the parabola to the focus is equal to its distance from the directrix.

### Step-by-Step Solution:

1. **Identify the Given Information:**
   - **Vertex**: \( (5, 6.5) \)
   - **Directrix**: \( y = 7.5 \) (a horizontal line)

2. **Understand the Parabola's Orientation:**
   - Since the directrix is a horizontal line, the parabola opens either upwards or downwards.
   - The vertex \( (5, 6.5) \) is below the directrix \( y = 7.5 \), so the parabola opens downwards.

3. **Determine the Distance Between the Vertex and the Directrix:**
   - The distance \( p \) from the vertex to the directrix is the absolute value of the difference in their \( y \)-coordinates.
   \[
   p = |7.5 - 6.5| = 1.
   \]

4. **Find the Focus:**
   - Since the parabola opens downwards and \( p = 1 \), the focus will be \( p \) units below the vertex.
   - The vertex is at \( (5, 6.5) \), so the focus will be at:
   \[
   \text{Focus} = (5, 6.5 - p) = (5, 6.5 - 1) = (5, 5.5).
   \]

### Conclusion:

- **Focus of the Parabola**: \( (5, 5.5) \)

The focus is located 1 unit below the vertex, which corresponds with the parabola opening downward as expected.

Would you like to explore any other properties of the parabola or need further explanations?

### Further Questions to Explore:

1. How do you find the equation of a parabola given its focus and directrix?
2. What are the steps to derive the parabola's equation from its vertex and focus?
3. How does changing the directrix affect the location of the focus?
4. What is the reflective property of a parabola, and how is it used in real-life applications?
5. How would you find the equation of the tangent line to the parabola at the vertex?

**Tip:** When dealing with parabolas, always verify the parabola's direction of opening to correctly determine the focus's location relative to the vertex.

Learn how to find the focus of a parabola given its vertex at (5, 6.5) and a directrix of y = 7.5. Understand the relationship between the vertex, directrix, and focus in parabolas. Suitable for high school students studying geometry.

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