To identify the focus and directrix of the parabola given by the equation \((y - 3)^2 = -4(x - 5)\), we will first rewrite this equation in standard form and identify the key components.
The given equation \((y - 3)^2 = -4(x - 5)\) is of the form \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus and also to the directrix, but in opposite directions.
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Identify \(h\) and \(k\):
- From the given equation, we can see that:
- \(k = 3\)
- \(h = 5\)
Thus, the vertex of the parabola is at the point \((5, 3)\).
- From the given equation, we can see that:
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Identify \(4p\):
- From the equation \((y - 3)^2 = -4(x - 5)\), we see:
- \(4p = -4\)
- Therefore, \(p = -1\).
- From the equation \((y - 3)^2 = -4(x - 5)\), we see:
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Determine the focus:
- Since this is a leftward-opening parabola (as indicated by the negative sign), the focus is located \(p\) units to the left of the vertex.
- The coordinates of the focus are: \[ (h + p, k) = (5 - 1, 3) = (4, 3) \]
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Determine the directrix:
- The directrix for a leftward-opening parabola is a vertical line located \(p\) units to the right of the vertex.
- Therefore, the directrix is given by the equation: \[ x = h + p = 5 + (-1) = 4 \]
Putting these results together, we find:
- Focus: \((4, 3)\)
- Directrix: \(x = 4\)
Thus, the correct response is: focus: \((4, 3)\); directrix: \(x = 4\).