To find the equation of the parabola with a focus at \((6, -5)\) and a directrix of \(x = 2\), we will follow the standard definition and properties of a parabola.
Step 1: Identify the Properties
- Focus: The focus of the parabola is at the point \((6, -5)\).
- Directrix: The directrix of the parabola is the vertical line \(x = 2\).
- Vertex: The vertex of the parabola is located halfway between the focus and the directrix.
Step 2: Calculate the Vertex
The vertex \(V\) is the average of the x-coordinates of the focus and the directrix line.
- x-coordinate of the focus: \(6\)
- Equation of the directrix: \(x = 2\)
The vertex \(x\) coordinate is calculated as follows: \[ x_V = \frac{6 + 2}{2} = \frac{8}{2} = 4 \] The y-coordinate of the vertex is the same as the y-coordinate of the focus since the focus is not vertically aligned with the directrix. Therefore, \[ y_V = -5 \] So the vertex is \((4, -5)\).
Step 3: Determine the Orientation
Since the directrix is a vertical line, the parabola will open to the right or to the left. Since the focus \((6, -5)\) is to the right of the directrix \(x = 2\), the parabola opens to the right.
Step 4: Find the Distance \(p\)
The distance \(p\) is the distance from the vertex to the focus (or from the vertex to the directrix). We can find this distance by calculating:
\[ p = 6 - 4 = 2 \]
Step 5: Write the Standard Form of the Parabola
The standard form of the equation of a horizontally oriented parabola (opening to the right) with vertex \((h, k)\) is: \[ (y - k)^2 = 4p(x - h) \] Where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.
Plugging the vertex \((4, -5)\) and \(p = 2\) into the formula:
- \(h = 4\)
- \(k = -5\)
- \(4p = 4 \cdot 2 = 8\)
Step 6: Write the Equation of the Parabola
The equation of the parabola becomes: \[ (y + 5)^2 = 8(x - 4) \]
Thus, the final equation of the parabola is: \[ \boxed{(y + 5)^2 = 8(x - 4)} \]