To write the equation of the parabola, we need to analyze the information given regarding the focus and directrix.
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Identify the Characteristics of the Parabola:
- The focus of the parabola is located at \( (10, -3) \).
- The directrix is a vertical line defined by the equation \( x = 2 \).
- Since the focus is to the right of the directrix, the parabola opens to the right.
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Determine the Vertex:
- The vertex of the parabola is located halfway between the focus and the directrix.
- The x-coordinate of the vertex can be determined by averaging the x-coordinates of the focus and the line \( x = 2 \): \[ \text{Vertex (h)} = \frac{10 + 2}{2} = \frac{12}{2} = 6. \]
- The y-coordinate of the vertex is the same as the y-coordinate of the focus since the directrix is vertical. Therefore, the y-coordinate of the vertex is \( -3 \).
So, the vertex is at \( (6, -3) \).
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Determine the Distance \( p \):
- The distance \( p \) is the distance from the vertex to the focus (or to the directrix).
- Since the focus is at \( (10, -3) \) and the vertex is at \( (6, -3) \), \[ p = 10 - 6 = 4. \]
- This means the parabola opens to the right and \( p = 4 \).
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Write the Equation of the Parabola:
- The general form of the equation of a horizontally opening parabola with vertex at \( (h, k) \) is given by: \[ (y - k)^2 = 4p(x - h). \]
- Substituting \( h = 6 \), \( k = -3 \), and \( p = 4 \): \[ (y + 3)^2 = 16(x - 6). \]
Thus, the equation of the parabola is: \[ (y + 3)^2 = 16(x - 6). \]
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