Compute the focal length of the parabola y^2  + 8x - 6y + 25 = 0.

To compute the focal length of the parabola given the equation \( y^2 + 8x - 6y + 25 = 0 \), we first need to rewrite the equation in the standard form of a parabola.

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

The given equation is:

\[
y^2 + 8x - 6y + 25 = 0
\]

First, group the terms involving \(y\):

\[
y^2 - 6y + 8x + 25 = 0
\]

Next, move the \(x\) and constant terms to the right side:

\[
y^2 - 6y = -8x - 25
\]

### Step 2: Complete the square

To complete the square on the left side for \(y\), we take the coefficient of \(y\), divide it by 2, and square it:

\[
\text{Coefficient of } y = -6
\]
\[
\left(\frac{-6}{2}\right)^2 = 9
\]

Add 9 to both sides of the equation:

\[
y^2 - 6y + 9 = -8x - 25 + 9
\]

This simplifies to:

\[
(y - 3)^2 = -8x - 16
\]

Factor out the right side:

\[
(y - 3)^2 = -8(x + 2)
\]

Now, the equation is in the standard form of a horizontal parabola:

\[
(y - k)^2 = 4p(x - h)
\]

Where \( (h, k) \) is the vertex of the parabola and \( p \) is the focal length.

### Step 3: Identify the focal length \(p\)

From the equation \( (y - 3)^2 = -8(x + 2) \), we can see that \( 4p = -8 \). Therefore:

\[
p = \frac{-8}{4} = -2
\]

The negative sign indicates that the parabola opens to the left. The absolute value of \(p\) is the focal length:

\[
\text{Focal length} = |p| = 2
\]

So, the focal length of the parabola is 2 units.

Would you like further details on this solution, or do you have any questions?

### Related Questions:
1. How do we find the vertex of a parabola given in general form?
2. What is the significance of the sign of \(p\) in the equation of a parabola?
3. How would the equation change if the parabola opened to the right instead?
4. Can you find the focus and directrix of the parabola from the standard form equation?
5. How do you derive the focal length for a vertically oriented parabola?

### Tip:
When completing the square, make sure to balance the equation by adding the same value to both sides. This step is crucial for converting a quadratic equation into its standard form.

Learn how to compute the focal length of the parabola y^2 + 8x - 6y + 25 = 0 with a step-by-step solution. This problem involves quadratic equations and parabolas, and includes completing the square to find the standard form. Suitable for students in Grades 10-12.

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