Question

What is the vertex form of the equation of the parabola with a focus at (52,48) and a directrix of `y=47`?

Answer 1

`y = (1/2)(x - 52)^2 + 47.5`

Explanation:

The vertex form of the equation of a parabola is:

`y = a(x - h)^2 + k` where (h, k) is the vertex point.

We know that the vertex is equidistant between the focus and the directrix, therefore, we split the distance between 47 and 48 to find that y coordinate of the vertex 47.5. We know that the x coordinate is the same as the x coordinate of the focus, 52. Therefore, the vertex is `(52, 47.5)`.

Also, we know that

`a = 1/(4f)` where `f` is the distance from the vertex to the focus:

From 47.5 to 48 is a positive `1/2`, therefore, `f = 1/2` thereby making `a = 1/2`

Substitute this information into the general form:

`y = (1/2)(x - 52)^2 + 47.5`