Question

the focus and directrix of a parabola are shown. what is the equation of the parabola?\
\
options:\
$y = -8(x - 2)^2 + 1$\
$y = \frac{1}{8}(x - 2)^2 - 1$\
$x = -\frac{1}{8}(y - 2)^2 - 1$\
$x = \frac{1}{8}(y - 2)^2 - 1$

Explanation:

Step1: Determine the vertex

The focus is \((1, 2)\) and the directrix is \(x = - 3\). The vertex \((h,k)\) is the midpoint between the focus and the directrix. For the \(x\)-coordinate of the vertex: \(h=\frac{1 + (-3)}{2}=\frac{-2}{2}=-1\), and the \(y\)-coordinate \(k = 2\) (since the parabola opens horizontally, the \(y\)-coordinate of the vertex is the same as the focus). So the vertex is \((-1,2)\).

Step2: Determine the direction and \(p\) value

Since the directrix \(x=-3\) is to the left of the focus \((1,2)\), the parabola opens to the right. The distance between the vertex and the focus (or directrix) is \(p\). The distance between \(x=-1\) (vertex \(x\)-coordinate) and \(x = 1\) (focus \(x\)-coordinate) is \(p=1-(-1)=2\).

Step3: Use the standard form of horizontal parabola

The standard form of a horizontal parabola is \(x=\frac{1}{4p}(y - k)^{2}+h\). Substitute \(h=-1\), \(k = 2\), and \(p = 2\) into the formula. First, calculate \(4p=4\times2 = 8\). Then the equation becomes \(x=\frac{1}{8}(y - 2)^{2}-1\).

Answer:

\(x=\frac{1}{8}(y - 2)^{2}-1\) (corresponding to the option \(x=\frac{1}{8}(y - 2)^{2}-1\) in the table)