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6 November, 17:45

Which is the equation of a parabola with a directrix at y = - 3 and a focus at (5, 3) ?

y = 1/12 (x-5) ^2

y = - 1/12 (x-5) ^2

y = 1/12 (x+5) ^2

y = - 1/12 (x--+5) ^2

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  1. 6 November, 18:05
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    y = 1/12 (x - 5) ²

    Step-by-step explanation:

    We can solve this graphically without doing calculations.

    The y component of the focus is y = 3. Since this is above the directrix, we know this is an upward facing parabola, so it must have a positive coefficient. That narrows the possible answers to A and C.

    The x component of the focus is x = 5. Since this is above the vertex, we know the x component of the vertex is also x = 5.

    So the answer is A. y = 1/12 (x-5) ².

    But let's say this wasn't a multiple choice question and we needed to do calculations. The equation of a parabola is:

    y = 1 / (4p) (x - h) ² + k

    where (h, k) is the vertex and p is the distance from the vertex to the focus.

    The vertex is halfway between the focus and the directrix. So p is half the difference of the y components:

    p = (3 - (-3)) / 2

    p = 3

    k, the y component of the vertex, is the average:

    k = (3 + (-3)) / 2

    k = 0

    And h, the x component of the vertex, is the same as the focus:

    h = 5

    So:

    y = 1 / (4*3) (x - 5) ² + 0

    y = 1/12 (x - 5) ²
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