Given the focus \( (6, -5) \) and the directrix \( x = 2 \), we can find the equation of the parabola as follows:
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Identify the vertex: The vertex is the midpoint between the focus and the directrix. Since the directrix is a vertical line, we calculate the x-coordinate of the vertex:
\[ x_v = \frac{x_{focus} + x_{directrix}}{2} = \frac{6 + 2}{2} = \frac{8}{2} = 4 \]
The y-coordinate of the vertex is the same as the y-coordinate of the focus:
\[ y_v = -5 \]
Thus, the vertex \( V \) is \( (4, -5) \).
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Calculate the distance \( p \): The distance \( p \) is the distance from the vertex to the focus (or to the directrix). Since the focus is to the right of the vertex:
\[ p = x_{focus} - x_v = 6 - 4 = 2 \]
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Determine the orientation: Since the focus is to the right of the directrix, the parabola opens to the right.
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Write the standard form: The standard form of a parabola that opens to the right is given by:
\[ (y - k)^2 = 4p(x - h) \]
where \( (h, k) \) is the vertex \( (4, -5) \) and \( p = 2 \).
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Substituting values: Substituting \( h = 4 \), \( k = -5 \), and \( p = 2 \):
\[ (y + 5)^2 = 4 \cdot 2 \cdot (x - 4) \]
\[ (y + 5)^2 = 8(x - 4) \]
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Final equation: The equation of the parabola is
\[ \boxed{(y + 5)^2 = 8(x - 4)} \]