Find the Diretix of a parabola if the focus is (-3,5) and a point on the parabola is (1,1)

Question

Help On geomtetry test  the answer is y= -4 right?

Answer 1

Since the directrix is parallel to the x-axis, and is greater than the focus, the parabola opens to the left and the vertex, (h, k), will be half way between them.

Fx = 3
x = 5

h = (Fx + x) / 2
h = (3 + 5) / 2
h = 8 / 2
h = 4

k = Fy
k = 1

The directrix crosses the x-axis at (5, 0), so

y = 0

and

x = a(y - k)² + h
5 = a(0 - 1)² + 4
5 = a + 4
a = 5 - 4
a = 1

x = (y - 1)² + 4
x = (y² - 2y + 1) + 4
x = y² - 2y + 1 + 4
x = y² - 2y + 5
¯¯¯¯¯¯¯¯¯¯¯¯¯

Answer 2

y = 5 is a horizontal line, so the parabola is vertical.

The directrix lies above the focus, so the parabola opens downwards.

The vertex of the parabola is halfway between the focus and directrix, at (3, 2).

p = distance between focus and vertex = 3
a = -1/(4p) = -1/12
y = (1/12)(x - 3)² + 2