Question

Ok so I have the question and answer but need to be explained why the answer works.

An archway will be constructed over a walkway. A piece of wood will need to be curved to match a parabola. Explain to Maurice how to find the equation of the parabola given the focal point and the directrix.

draw a perpendicular line from the directrix passing through the focus, this will be the line of symmetry. The vertex(h, k) will be located on the line half way between the focus and directrix. The distance from the focus to the vertex is called the focal length, call it a. The then equation is (x - h)^2 = 4a(y - k) the equation can be manipulated to y = 1/4a(x - h)^2 + k

If someone could reword this answer to make it simpler, that would be great.

Answer 1

as you know the directrix and focus. so first of all find the distance 'a.f' shown in the figure. now find the location of the mid point of the line 'a.f' because it's the vertex. half the distance to get the focal length 'a'. now put all the values in the equation given to get the equation of required parabola.