To determine the focus and directrix of the parabola, we first need to rewrite the given equation in the standard form of a parabola. The given equation is:

$x^2 - 6x - 2y + 9 = 0$

Step 1: Complete the Square

Start by isolating the $x$ terms on one side of the equation:

$x^2 - 6x = 2y - 9$

Now, complete the square on the left side. To do this, take half the coefficient of $x$, square it, and add it to both sides of the equation. The coefficient of $x$ is $-6$, so:

$\left(\frac{-6}{2}\right)^2 = 9$

Add 9 to both sides:

$x^2 - 6x + 9 = 2y - 9 + 9$

This simplifies to:

$(x - 3)^2 = 2y$

Step 2: Convert to Standard Form

Now, the equation is in the standard form of a parabola that opens upwards or downwards:

$(x - h)^2 = 4p(y - k)$

where $(h, k)$ is the vertex, and $p$ is the distance from the vertex to the focus and directrix. Comparing with the standard form:

$(x - 3)^2 = 4\left(\frac{1}{2}\right)(y - 0)$

Here, $h = 3$, $k = 0$, and $4p = 2$, so $p = \frac{1}{2}$.

Step 3: Determine the Focus and Directrix

• Vertex: $(h, k) = (3, 0)$
• Focus: The focus lies $p$ units above the vertex, since the parabola opens upwards. Thus, the focus is at $(3, \frac{1}{2})$.
• Directrix: The directrix is a horizontal line $p$ units below the vertex, so the directrix is $y = -\frac{1}{2}$.

Step 4: Sketch the Graph

• The vertex is at $(3, 0)$.
• The focus is at $(3, \frac{1}{2})$.
• The directrix is the line $y = -\frac{1}{2}$.

Let me know if you want a sketch of the graph or if you have any questions.

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Here are some related questions you might find useful:

1. How would you determine the vertex, focus, and directrix for a parabola that opens sideways?
2. What are the differences in the equations of parabolas that open up, down, left, and right?
3. How can you derive the equation of a parabola given the vertex and focus?
4. How does changing the value of $p$ affect the shape of the parabola?
5. What is the significance of the directrix in relation to the parabola's shape?

Tip: Completing the square is a useful technique not just for conic sections but also for solving quadratic equations and optimization problems.