Question

HELP ASAP In two or more complete sentences, Explain how you would find the equation of a parabola, given the coordinate of the focus and the equation of the directrix. any help would be wonderful even if it’s a paragraph long or not even a sentence at all any help at all would be greatly appreciated

Answer 1

In the general case in Cartesian coordinates, you would use the definition of a parabola as the locus of points equidistant from the focus and directrix. The equation would equate the square of the distance from a general point (x, y) to the focus with the square of the distance from that point to the directrix line.

Suppose the focus is located at (h, k) and the equation of the directrix is ax+by+c=0. The expression for the square of the distance from (x, y) to the point (h, k) is ...
  (d₁)² = (x-h)²+(y-k)²
The expression for the square of the distance from (x, y) to the directrix line is
  (d₂)² = (ax+by+c)²/(a²+b²)

Equating these expressions gives the equation of the parabola.
  (x-h)²+(y-k)² = (ax+by+c)²/(a²+b²)

When the directrix is parallel with one of the axes, one of the coefficents "a" or "b" is zero and the equation becomes much simpler. Often, it would be easier to make use of the formula (for a directrix parallel to the x-axis):
  y = 1/(4p)*(x -h)² +k
where the (h, k) here is the vertex, the point halfway between the focus and directrix, and "p" is the (signed) distance from the focus to the vertex. (p is positive when the focus is above or to the right of the vertex.)