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The image contains a question about a parabola defined by the equation:

\[
x^2 + 6x + 12y - 15 = 0
\]

The task is to determine various properties of the parabola such as:
- **Vertex** (Titik Puncak)
- **Focus** (Titik Fokus)
- **Equation of the directrix** (Persamaan garis direktriks)
- **Check if the point (1,2) lies on the parabola**
- **Length of the latus rectum (Panjang Latus Rectum)**

### Let's break down the equation and solve it step-by-step.

1. **Rewrite the equation in vertex form** by completing the square.

Starting with the given equation:

\[
x^2 + 6x + 12y - 15 = 0
\]

Group the terms involving \(x\):

\[
x^2 + 6x = -12y + 15
\]

Complete the square on the left-hand side:

\[
x^2 + 6x + 9 = -12y + 15 + 9
\]

\[
(x + 3)^2 = -12y + 24
\]

\[
(x + 3)^2 = -12(y - 2)
\]

Now, we have the equation of the parabola in the standard form:

\[
(x + 3)^2 = -12(y - 2)
\]

This is the equation of a **parabola opening downwards**.

2. **Find the vertex** (Titik Puncak).

From the equation \((x + 3)^2 = -12(y - 2)\), the vertex \((h, k)\) is \((-3, 2)\).

So, the vertex is:

\[
Titik Puncak: (-3, 2)
\]

3. **Find the focus** (Titik Fokus).

The standard form for a vertical parabola is \((x - h)^2 = 4p(y - k)\), where \(p\) is the distance from the vertex to the focus. In this case, \(4p = -12\), so \(p = -3\).

The focus is \(3\) units below the vertex, so:

\[
\text{Focus} = (-3, 2 - 3) = (-3, -1)
\]

4. **Find the equation of the directrix** (Persamaan garis direktriks).

The directrix is a horizontal line \(p\) units above the vertex. Since \(p = -3\), the directrix is \(y = 2 + 3 = 5\).

So, the equation of the directrix is:

\[
y = 5
\]

5. **Check if the point (1,2) lies on the parabola**.

Substitute \(x = 1\) and \(y = 2\) into the original equation \(x^2 + 6x + 12y - 15 = 0\):

\[
1^2 + 6(1) + 12(2) - 15 = 1 + 6 + 24 - 15 = 16
\]

Since \(16 \neq 0\), the point \((1,2)\) does **not** lie on the parabola.

6. **Find the length of the latus rectum** (Panjang Latus Rectum).

The length of the latus rectum is given by \(|4p|\). Since \(4p = 12\), the length is:

\[
\text{Latus Rectum} = 12 \text{ units}
\]

Would you like further explanation on any of the steps? 

Here are 5 related questions for practice:
1. How do you find the vertex of a parabola from its equation?
2. What is the significance of the directrix in a parabola?
3. How do you determine if a point lies on a parabola?
4. Can the value of \(p\) be negative, and what does it signify?
5. How is the length of the latus rectum calculated?

**Tip**: Completing the square is a very useful technique for converting quadratic equations into a more understandable form, such as vertex form.

Diberikan persamaan parabola x^2 + 6x + 12y - 15 = 0. Temukan Titik Puncak, Titik Fokus, Persamaan Garis Direktriks, apakah titik (1,2) terletak pada parabola, dan panjang Latus Rectum.

Solve the parabola equation x^2 + 6x + 12y - 15 and learn how to find the vertex, focus, directrix, and the length of the latus rectum. Understand whether the point (1,2) lies on the parabola. Suitable for Grades 9-12 students.

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