16. Find the equation of a parabola with a focus at (3,1) and a directrix at y = 3. A) y = 1∕4(x – 3)^2 + 3
B) y = –1∕4 (x – 3)^2 + 3
C) y = –1∕4 (x – 3)^2 + 2
D) y = 1∕4(x – 3)^2 + 2

C

Step-by-step explanation:

Any point (x, y) on the parabola is equidistant from the focus and the directrix.

Using the distance formula

= | y - 3 | ← square both sides

(x - 3)² + (y - 1)² = (y - 3)² ← expand the y- factors

(x - 3)² + y² - 2y + 1 = y² - 6y + 9 ← subtract y² - 2y + 1 from both sides

(x - 3)² = - 4y + 8 ( subtract 8 from both sides )

(x - 3)² - 8 = - 4y ( divide both sides by - 4 )

- (x - 3)² + 2 = y, that is

y = - (x - 3)² + 2 → C