

1. Give the focus and directrix of the parabola with equation x ^ 2 = 10y



The given equation of the parabola is:

\[
x^2 = 10y
\]

This is a vertical parabola that opens upwards since it is in the form \(x^2 = 4py\), where \(p\) is the distance between the vertex and the focus (and also the distance between the vertex and the directrix, but in the opposite direction).

### Step 1: Identify the value of \(p\)

In the standard form \(x^2 = 4py\), we compare this with the given equation \(x^2 = 10y\).

Thus, \(4p = 10\), so:

\[
p = \frac{10}{4} = 2.5
\]

### Step 2: Find the focus and directrix

- The vertex of the parabola is at the origin \((0, 0)\).
- The focus is located at \((0, p)\), which is \((0, 2.5)\).
- The directrix is the line \(y = -p\), which is \(y = -2.5\).

### Conclusion

- **Focus**: \((0, 2.5)\)
- **Directrix**: \(y = -2.5\)

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What is the equation of a parabola with a horizontal axis of symmetry?
2. How do you derive the equation of a parabola from its focus and directrix?
3. What is the relationship between the vertex, focus, and directrix in a parabola?
4. How would the focus and directrix change if the parabola opened downward?
5. How do you convert a parabola's equation from standard form to vertex form?

**Tip:** The value of \(p\) in a parabola equation determines how "stretched" or "narrow" the parabola appears.

Give the focus and directrix of the parabola with equation x^2 = 10y

Explore how to find the focus and directrix of the parabola with equation x^2 = 10y. This solution breaks down the key steps, including identifying the value of p, the focus at (0, 2.5), and the directrix y = -2.5. Perfect for students in Grades 9-11 studying coordinate geometry and conic sections.

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