Question

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

Answer 1

if you take any point (x,y) on the curve the distance between the focus and that point = the perpendicular distance between the point and the directrix (definition of a parabola)

first distance = sqrt[ (y - (-5))^2 + (x - (-5))^2] = sqrt [ (x +5)^2 + (y + 5)^2]
and second distance is y - 7

Answer 2

so we have
sqrt [ (x +5)^2 + (y + 5)^2] = (y - 7

square both sides
(x +5)^2 + (y + 5)^2 = (y - 7)^2

simplify this and you'll get the required equation