Answer:
The answer is below
Step-by-step explanation:
A parabola is the locus of a point such that its distance from a given line (directrix) and a fixed point (focus) is always the same.
The equation of a parabola with a horizontal line as directrix is given by:
(x - h)² = 4p(y - k)
where y = k - p is the directrix and the focus = (h, k + p)
Given that parabola has a directrix at y = 2 and a focus at (5, 0)
Therefore:
y = k - p
k - p = 2 (1)
focus = (h, k + p) = (5, 0)
h = 5, k + p = 0
k + p = 0 (2)
Solve equation 1 and 2 simultaneously by adding both equations together:
2k = 2
k = 1
Substitute k = 1 in k + p = 0 to find p:
1 + p = 0
p = -1
Therefore h = 5, p = -1 and k = 1
Substitute the values of h, k and p into the equation of a parabola:
(x - h)² = 4p(y - k)
(x - 5)² = 4(-1)(y - 1)
(x - 5)² = -4(y - 1)
x² - 10x + 25 = -4y + 4
-4y = x² - 10x + 21
4y = 10x - x² - 21
