Question

given a parabola with focus (2, 6) and directrix y = 4, which of the following represents the correct equation of this parabola?

$y=\frac{1}{4}(x - 2)^2-5$

$y=\frac{1}{4}(x + 2)^2+5$

$y=\frac{1}{2}(x - 2)^2+5$

$y=\frac{1}{4}(x - 2)^2+5$

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus $(2,6)$ and the point on the directrix $y = 4$ directly below (or above) the focus. The $x$ - coordinate of the vertex is the same as the $x$ - coordinate of the focus, $x=2$. The $y$ - coordinate of the vertex is $\frac{6 + 4}{2}=5$, so the vertex is $(h,k)=(2,5)$.

Step2: Determine the value of $a$

The distance $a$ between the vertex and the focus (or the vertex and the directrix) is $a=\frac{6 - 5}{1}=1$. For a parabola of the form $y=a(x - h)^2+k$, substituting $h = 2$, $k = 5$ and $a=\frac{1}{4}$ (since the general formula for the distance from the vertex to the focus for $y=a(x - h)^2+k$ is $|a|=\frac{1}{4p}$, and here $p = 1$), we get $y=\frac{1}{4}(x - 2)^2+5$.

Answer:

$y=\frac{1}{4}(x - 2)^2+5$ (the last option in the multiple - choice list)