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We have two equations representing parabolas, and the task is to find the vertex, focal distance, and latus rectum for each.

### 1. \( x^2 = 12y \)

This is the equation of a **vertical parabola** of the form \( x^2 = 4py \), where \( p \) is the distance from the vertex to the focus (focal distance).

#### Step-by-step:
- **Vertex**: The standard form is \( x^2 = 4py \), which means the vertex is at \( (0, 0) \).
- **Focal distance (p)**: From the equation \( x^2 = 12y \), we can compare it to \( x^2 = 4py \), so \( 4p = 12 \), giving \( p = 3 \).
- **Focus**: The focus is at \( (0, p) \), which is \( (0, 3) \).
- **Directrix**: The directrix is at \( y = -p \), so \( y = -3 \).
- **Latus Rectum**: The latus rectum is \( |4p| \), so \( |4(3)| = 12 \).

### 2. \( y^2 = 12x \)

This is the equation of a **horizontal parabola** of the form \( y^2 = 4px \).

#### Step-by-step:
- **Vertex**: Similarly, the vertex is at \( (0, 0) \).
- **Focal distance (p)**: From the equation \( y^2 = 12x \), we compare it to \( y^2 = 4px \), so \( 4p = 12 \), giving \( p = 3 \).
- **Focus**: The focus is at \( (p, 0) \), which is \( (3, 0) \).
- **Directrix**: The directrix is at \( x = -p \), so \( x = -3 \).
- **Latus Rectum**: The latus rectum is \( |4p| \), so \( 12 \) (same as for the vertical parabola).

### Summary:
1. **Equation** \( x^2 = 12y \):
   - Vertex: \( (0, 0) \)
   - Focal distance: \( 3 \)
   - Focus: \( (0, 3) \)
   - Directrix: \( y = -3 \)
   - Latus rectum: \( 12 \)

2. **Equation** \( y^2 = 12x \):
   - Vertex: \( (0, 0) \)
   - Focal distance: \( 3 \)
   - Focus: \( (3, 0) \)
   - Directrix: \( x = -3 \)
   - Latus rectum: \( 12 \)

Would you like more details on these steps or have any specific questions?

### Related questions:
1. How do we derive the focus of a parabola from its equation?
2. What is the geometric significance of the directrix in a parabola?
3. How does the orientation of a parabola change based on its equation?
4. What other forms can a parabola take besides horizontal and vertical?
5. How is the latus rectum connected to the properties of a parabola?

### Tip:
In general, the latus rectum always equals the absolute value of \( 4p \) for any parabola. It helps visualize the width of the parabola near its focus!

Determine the vertex, focal distance and the latus rectum for 1. x^2 = 12y 2. y^2 = 12x

Learn how to determine the vertex, focal distance, and latus rectum for two parabolas: x^2 = 12y (vertical parabola) and y^2 = 12x (horizontal parabola). This problem explores key conic section concepts suitable for students in Grades 10-12.

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