Question

The focus of a parabola is (-4, -5), and its directrix is y = -1. Fill in the missing terms and signs in the parabola's equation in standard form

Answer 1

We are given the focus and directrix of a parabola:

Focus (-4, -5)
Directrix = y = -1

The standard equation of a parabola with vertex (h, k) is

(y-k) = 4a (x-h)^2

and

focus: (h +a, k)
directrix: y = h - a

Now, we have k = -5

and
h +a = -4
h - a = -1

solve for h and a

a = h +1
h + h + 1 = -4
h = -3/2
a = -1/2

Therefore the vertex of the parabola is (-1.5, -5)

Answer 2

(x+4)^2 = -8(y+3)

Explanation:

Given that the focus of a parabola is (-4, -5), and its directrix is y = -1.

We know that a parabola is a curve which has equal distance from focus and the directrix

If (x,y) be any point on the parabola, then

`√((x+4)^2+(y+5)^2) =y+1\\`

Square both the sides

`(x+4)^2+(y+5)^2=(y+1)^2\\(x+4)^2 = (2y+6)(-4)\\(x+4)^2 = -8(y+3)`

Thus std form of the parabola is

(x+4)^2 = -8(y+3)