Final answer:
The equation of a parabola with a vertex (-2, 3) and directrix x = 6 should be of the form x = a(y - k)² + h, since it opens horizontally. Calculating 'a' as -1/32 and translating this into the standard form results in x = -(y - 3)² + 62. Given options do not match this equation exactly but the closest option is d) x = -(y - 3)² + 2.
Step-by-step explanation:
The question involves finding the equation of a parabola with a given vertex and directrix. In this case, the vertex is (-2, 3) and the directrix is x = 6. Since the directrix is a vertical line, we know that the parabola opens either left or right, not up or down. For parabolas with vertical directrices, the standard form of the equation is x = a(y - k)² + h, where (h, k) is the vertex. The 'a' value determines the direction and width of the parabola.
Since the directrix is to the right of the vertex, our parabola opens to the left, which means 'a' is negative. Moreover, the distance from the vertex to the directrix is the absolute value of -2 (the x-coordinate of the vertex) minus 6 (the x-coordinate of the directrix), which is |-2 - 6| = 8. This distance is also 1/(4a), so 1/(4a) = 8, meaning a = -1/32. Therefore, our equation becomes x = -(y - 3)²/32 - 2. After multiplying by 32 to clear the fraction, we get x = -(y - 3)² + 64 - 2, simplifying to x = -(y - 3)² + 62.
Since none of the options precisely match this form, there may be an error within the given options. However, the most similar option in form is option d) x = -(y - 3)² + 2. Note that if we assume that the original question meant 2 instead of 62, then option d would indeed be the mention correct option.
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