To write the equation of a parabola given its focus and directrix, we can use the definition of a parabola. A parabola is the set of all points (x, y) that are equidistant from a fixed point called the focus and a fixed line called the directrix.
Assume that the focus is at the point (h, k) and the directrix is the line represented by the equation y = d, where d is a constant value (for vertical parabolas) or x = d (for horizontal parabolas).
For vertical parabolas, if the focus is at (h, k) and the directrix is y = d, the distance from any point (x, y) on the parabola to the focus equals the distance from (x, y) to the directrix. This leads us to the equation:
(x – h)2 = 4p(y – k)
where p = k – d. This means if the focus is above the directrix, p is positive; if below, p is negative.
For horizontal parabolas with the focus at (h, k) and the directrix x = d, the corresponding equation would be:
(y – k)2 = 4p(x – h)
where p = h – d.
In summary, determine the coordinates of the focus and the equation of the directrix, then apply the respective formula to obtain the equation of the parabola.