Refer to the diagram shown below.
At an arbitrary point P (x,y) on the parabola, the distance, d, from the focus to P should be equal to the distance from P to the directrix.
That is,
d² = (x - 3)² + (y - 4)² = (y - 1)²
(x - 3)² + y² - 8y + 16 = y² - 2y + 1
(x - 3)² + 15 = 6y
Divide each side by 6 to obtain
y = (1/6) (x - 3)² + 5/2
The vertex of the parabola is at (3, 5/2).
The line of symmetry is x = 3.
The leading coefficient, 1/6, is positive, therefore the curve opens upward.
Answer: [tex]y= \frac{1}{6}(x-3)^{2}+ \frac{5}{2} [/tex]
