Answer:
D. y = 3·x² - 2·x + 11/12
Step-by-step explanation:
The coordinates of the focus of the parabola, f = (1/3, 2/3)
The directrix of the parabola, y = 1/2
The standard form of the equation of a parabola is (x - h)² = 4·p·(y - k)
The coordinates focus = (h, k + p)
The y-coordinates of the directrix, y = k - p
By comparison, we have;
h = 1/3...(1)
k + p = 2/3...(2)
k - p = 1/2...(3)
Adding equation (3) to equation (2) gives;
k + p + (k - p) = k + k + p - p = 2·k = 2/3 + 1/2 = 7/6
k = (7/6)/2 = 7/12
k = 7/12
From equation (2), we get;
p = 2/3 - k
∴ p = 2/3 - 7/12 = 1/12
p = 1/12
The equation of the parabola is therefore;
(x - 1/3)² = 4·(1/12)·(y - 7/12) = y/3 - 7/36
y = 3 × ((x - 1/3)² + 7/36) = 3 × ((x² - 2·x/3 + 1/9) + 7/36) = 3·x² - 2·x + 11/12
y = 3·x² - 2·x + 11/12.
