Answer:
(y - 4)² = 12(x - 5)
Step-by-step explanation:
the vertex and focus lie on the principal axis y = 4
The focus is inside the parabola to the right of the vertex
This is therefore a horizontally opening parabola
with equation
(y - k)² = 4p(x - h)
where (h, k) are the coordinates of the vertex and p is the distance from the vertex to the focus/ directrix
here p = 8 - 5 = 3
The directrix is vertical and to the left of the vertex with equation x = 2
From any point (x, y) on the parabola the focus and directrix are equidistant
using the distance formula, then
[tex]\sqrt{(x-8)^2+(y-4)^2}[/tex] = | x - 2 |
squaring both sides
(x - 8)² + (y - 4)² = (x - 2)²
(y - 4)² = (x - 2)² - (x - 8)² = x² - 4x + 4 - x² + 16x - 64 = 12x - 60
(y - 4)² = 12(x - 5) ← equation of parabola
