*Find the focus and directrix of the parabola. x*2 +4*y* = 0

To find the focus and directrix of the parabola given by the equation:

\[
x^2 + 4y = 0
\]

### Step 1: Rewrite the equation
Reorganize it into the standard form of a parabola:

\[
x^2 = -4y
\]

This equation is in the form of:

\[
x^2 = 4py
\]

where \( p \) is a constant that determines the distance from the vertex to the focus and directrix.

### Step 2: Identify the value of \( p \)
In the equation \( x^2 = -4y \), we can see that:

\[
4p = -4 \implies p = -1
\]

### Step 3: Determine the vertex, focus, and directrix
- **Vertex**: The vertex of this parabola is at the origin \((0, 0)\).

- **Focus**: Since \( p = -1 \), the focus is 1 unit below the vertex (because the parabola opens downward along the y-axis). Thus, the focus is at:

\[
(0, -1)
\]

- **Directrix**: The directrix is a horizontal line \( p \) units above the vertex. Therefore, the equation of the directrix is:

\[
y = 1
\]

### Final Answer:
- **Focus**: \((0, -1)\)
- **Directrix**: \( y = 1 \)

Would you like more details on any part of this solution or have any questions?

Here are 5 related questions for further exploration:
1. How do you find the vertex of a parabola from its standard form?
2. What is the difference between parabolas that open upward and those that open downward?
3. How would you find the focus and directrix if the equation was \( y^2 = 4x \)?
4. What is the geometric interpretation of a parabola's focus and directrix?
5. How does changing the value of \( p \) affect the shape and position of a parabola?

**Tip**: When a parabola is in the form \( x^2 = 4py \), a positive \( p \) indicates the parabola opens upward, while a negative \( p \) indicates it opens downward.

Find the focus and directrix of the parabola. x^2 + 4y = 0

Learn how to find the focus and directrix of the parabola given by the equation x^2 + 4y = 0. This detailed step-by-step solution explains the key concepts of conic sections and provides insight into the standard form of a parabola. Suitable for students in Grades 10-12.

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