A parabola with focus at (2,5) and directrix y=-1

To find the equation of a parabola with a focus at \( (2, 5) \) and directrix \( y = -1 \), follow these steps:

1. **Determine the vertex**:
   The vertex lies midway between the focus and the directrix. The y-coordinate of the vertex is the average of the y-coordinate of the focus (5) and the directrix (-1):
   \[
   y_{\text{vertex}} = \frac{5 + (-1)}{2} = \frac{4}{2} = 2
   \]
   Since the x-coordinate of the focus is \( 2 \), the vertex is at \( (2, 2) \).

2. **Determine the distance between the focus and the directrix**:
   The distance between the focus \( (2, 5) \) and the directrix \( y = -1 \) is \( |5 - (-1)| = 6 \). Half of this distance is the focal length \( p \), which is:
   \[
   p = \frac{6}{2} = 3
   \]

3. **Equation of the parabola**:
   The standard form of a vertical parabola with vertex \( (h, k) \) and focal length \( p \) is:
   \[
   (x - h)^2 = 4p(y - k)
   \]
   Here, \( (h, k) = (2, 2) \) and \( p = 3 \). Substituting these values into the equation:
   \[
   (x - 2)^2 = 4(3)(y - 2)
   \]
   Simplifying:
   \[
   (x - 2)^2 = 12(y - 2)
   \]

Thus, the equation of the parabola is \( (x - 2)^2 = 12(y - 2) \).

Would you like further details or have any questions?

Here are five related questions:
1. How do we derive the equation of a horizontal parabola?
2. What is the significance of the directrix in a parabola?
3. How can we generalize the equation of a parabola with focus and directrix?
4. How does the focal length affect the shape of a parabola?
5. What is the role of the vertex in determining the direction of the parabola?

**Tip**: The focal length \( p \) helps in determining the "width" or "narrowness" of a parabola—the larger the value of \( p \), the wider the parabola opens.

Find the equation of a parabola with a focus at (2,5) and a directrix at y=-1. Learn how to calculate the vertex and the focal length to derive the equation (x - 2)^2 = 12(y - 2), covering key concepts of conic sections and parabolas. Suitable for students in Grades 10-12.

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