Find the verter, focus, eq of directrix, length of latusrectum y ^ 2 - 4y - 8x + 12 = 0 .

To find the vertex, focus, equation of the directrix, and length of the latus rectum for the given equation of a parabola:

\[
y^2 - 4y - 8x + 12 = 0
\]

we need to first rewrite it in the standard form of a parabola equation. Let's go through the steps:

### Step 1: Rewrite the equation in standard form

The given equation is:

\[
y^2 - 4y - 8x + 12 = 0
\]

We can rearrange it to isolate the \( y \)-terms:

\[
y^2 - 4y = 8x - 12
\]

### Step 2: Complete the square for the \( y \)-terms

To complete the square, take the coefficient of \( y \), divide by 2, and square it:

\[
\text{Coefficient of } y = -4, \quad \text{Half of } -4 = -2, \quad (-2)^2 = 4
\]

Now add and subtract 4 on the left-hand side:

\[
(y^2 - 4y + 4) - 4 = 8x - 12
\]

This can be rewritten as:

\[
(y - 2)^2 - 4 = 8x - 12
\]

Add 4 to both sides to simplify:

\[
(y - 2)^2 = 8x - 8
\]

Factor out the right-hand side:

\[
(y - 2)^2 = 8(x - 1)
\]

### Step 3: Identify the parameters

The equation \((y - k)^2 = 4p(x - h)\) represents a parabola that opens either left or right. Here:

- \( (h, k) = (1, 2) \) is the **vertex**.
- \( 4p = 8 \), so \( p = 2 \).

### Step 4: Find the focus

The focus of the parabola is at \( (h + p, k) \). Since \( p = 2 \):

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

### Step 5: Equation of the directrix

The directrix is a vertical line given by \( x = h - p \):

\[
x = 1 - 2 = -1
\]

### Step 6: Length of the latus rectum

The length of the latus rectum for a parabola is given by \( |4p| \). Here:

\[
\text{Length of latus rectum} = |4p| = 8
\]

### Final Answer:

- **Vertex**: \( (1, 2) \)
- **Focus**: \( (3, 2) \)
- **Equation of the directrix**: \( x = -1 \)
- **Length of the latus rectum**: \( 8 \)

Would you like further details on any of these steps? Here are 5 related questions:

1. How do you complete the square for a quadratic expression?
2. What is the general form of a parabola that opens horizontally?
3. How can we derive the focus and directrix for a parabola from its equation?
4. What are some applications of parabolas in real-world problems?
5. How do you find the vertex of a parabola using calculus?

**Tip:** Completing the square is a useful technique not just for parabolas, but also for solving quadratic equations and analyzing the properties of conic sections.

Learn how to find the vertex, focus, equation of the directrix, and length of the latus rectum for the parabola described by the equation y^2 - 4y - 8x + 12 = 0. This problem involves completing the square, identifying key parameters of a parabola, and applying formulas for focus, directrix, and latus rectum. Suitable for advanced high school students and above.

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