Derive the equation of the parabola with a focus at (-5,-5) and a directrix of y=7

Answer 1

desmos.com/calculator/f0skni0koz

Answer 2

vertex is midway between focus and directrix. -5 and 7 have a midway of 1. (7--5)/2 = 12/2 =6. 7-6= 1

vertex is (-5,1)

parabola equation is y-h= a(x-k)^2 with (k,h) = vertex = (-5,1)

We need on more point on the parabola to calculate "a" the coefficient of the x^2 term.

(-5,-5) is the focus. It's below the directrix and below the vertex. This is a downward sloping parabola with a <0. a is negative.

find another point on the parabola, just horizontally east of the focus, a point (x,-5) which is equidistant from the directrix. It's 12 from the directrix and 12 from the focus. that means the x in (x, -5) is 7 (7,-5) Plug that point into the equation y-1 = a(x+5)^2 to calculate "a"

-5 -1= a(7+5)^2

-6 = 144a

a= -6/144= -3/72

y-1 = (-3/72)(x+5)^2

72y- 72 = -3(x^2 +10x +25)

72y = -3x^2 -30x -75 + 72

72y = -3x^2 -30x -3