Derive the equation of the parabola with a focus at (−5, −5) and a directrix of y = 7. a. f(x) = −one twenty-fourth (x − 1)2 − 5 b. f(x) = one twenty-fourth (x − 1)2 − 5 c. f(x) = −one twenty-fourth (x + 5)2 + 1 d. f(x) = one twenty-fourth (x + 5)2 + 1

First find the distance between the focus and the point (x, y)

distance = sqrt ( (x + 5}^2 + (y + 5)^2)

distance between the directrix and (x, y) is

|y - 7|

These distances are equal  so , squaring we have:-

(x + 5)^2 + (y + 5)^2 = (y - 7)^2

x^2 + 10x + 25 + y^2 + 10y + 25 =  y^2 - 14y + 49

24y =  -x^2 - 10x - 1

24y =  - (x^2 + 10x) - 1

24y = - [(x + 5)^2 - 25] - 1

24y = -(x + 5)^2  + 24

y = -(1/24)(x + 5)^2 + 1       (answer)

Option C

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